MICROBIALFUELCELLS:DESIGN,CONTROL-ORIENTEDMODELING,AND EXPERIMENTALRESULTS By AliAbul ATHESIS Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof MechanicalEngineering-MasterofScience 2015 ABSTRACT MICROBIALFUELCELLS:DESIGN,CONTROL-ORIENTEDMODELING, ANDEXPERIMENTALRESULTS By AliAbul Thereisnodoubtabouthowcrucialtohavesustainableenergyinthisera.Researchers focusonfuelcellsbecauseoftheirhighency,environmentalfriendliness,andindepen- dencefromlimitedsources,etc.Microbialfuelcell(MFC)isapromisingtechnologythat respondstothedemandofsustainableenergy.MFCs,similartootherfuelcells,usecat- alystsandproduceelectricitythroughchemicalreactionsduringsubstratebreak-down.In thiscase,MFCsusebacteriaasthecatalyststobreakdowntheorganicmatter. Therehavebeencontrolstudiesonfuelcells,sponhydrogenfuelcells,forvarious purposes.BecauseMFCsarestillnotwellunderstood,similarcontrolstudieshavenot beenadequatelyconducted.Inthisstudy,acontrol-orientedmathematicalmodelforMFC dynamicsisdevelopedandanalyzed.AnMFCsystemisdesignedanddeveloped,whichhas successfullydemonstratedproductionofelectricity.Experimentsareconductedtoidentify themodelparametersandvalidatethemodel.FortheMFCprototype, G.sulfurreducens strainPCAisusedasthepurebacteriaculturewiththeacetateasthesubstrate.The MFCusedinthisstudyadoptsamembrane-lesssingle-chamberandutilizes anair-cathodeandacarbon-brushanode. Oncethemodelisdeveloped,thebehaviorofanMFCisanalyzedusingsystemtheory. Inparticular,theequilibriaofthesysteminthecontinuousmode,wheretheMFCisfed withthesubstrateataconstantratearecomputed.Furthermore,Jacobiananalysisand phaseportraitsareusedtounderstandthestabilitypropertiesoftheequilibria. TomydearestparentsZeynepandMehmetResitAbul,mylovelywifeStephanieAnn Duperon,mysupportivesiblingsBeratandMeralAbul. iii ACKNOWLEDGMENTS ThisresearchwassupportedinpartbytheNationalScienceFoundation(IIS1319602).I ammostgratefultomyadvisor,thedirectorofSmartMicrosystemsLabatMichiganState University,Prof.XiaoboTan.ThroughmyentirestudyProf.Tanguidedmewiselywith hisextensiveexperienceandknowledge.Prof.Tanencouragedmetobearesearcherbut alsopreparedmeforbecominganengineerwithtknowledgeandskillsfrommultiple disciplines.Hisvaluableguidanceandgenerousencouragementbmealotinmy researchandwillcontinuetobmyfuturecareer. Iwanttothankmycommitteemembers:Prof.HassanKhalilandProf.GuomingZhu atMichiganStateUniversity,forjoiningmyadvisorycommitteeandmeinsightful comments.Theirextensiveknowledgeandguidanceimprovedthequalityofmyresearch. Inthisstudy,Prof.GemmaRegueraandherlabmembersprovidedimmensehelpduring myprototypingandexperiments.IamspindebtedtoDr.RebeccaSteidl, whogavemeusefulcommentsonmysystemandmetmost(ifnotall)ofmyneedsinthe experimentalpartofthisstudy. Iamverygratefultomylabmembers.Webrainstormedforeachother'sendlessproblems andhadfunworkingtogether.SpecialthanksgotoJohnThon,OsamaEnnasr,HongLeiand JunZhang,whodirectlycontributedtomyresearch.Iwouldalsoliketothankmembers intheECEdepartment,especiallyBrianWright,GreggMulderandRoxannePeacock.Dr. RezaLoloeegenerouslyletmeusetheequipmentinhislabformymaterialsfabrication, whichwasgreatlyappreciated.IwanttogivespecialthankstoProf.BruceRittmann,Prof. BruceLogan,andProf.AndrewMarcuswhoansweredmyquestionspatientlythroughmy research. iv Iwouldliketothankmyfamily:myparents,mymother-in-law,myfather-in-law,my brother,andmysisterforsupportingmespirituallythroughoutmythesisresearchand throughoutmylifeingeneral.Yourprayerformewaswhatsustainedmethusfar. Lastbutnotleast,Iwouldliketogiveaveryspecialthank-youtomydearestwife,who supportedmethroughoutthisstudyaswell. v TABLEOFCONTENTS LISTOFTABLES .................................... viii LISTOFFIGURES ................................... ix Chapter1Introduction ................................ 1 1.1Background....................................1 1.2LiteratureReview.................................4 1.2.1FuelCells.................................4 1.2.2MicrobialFuelCells...........................7 1.2.2.1MFCWorkingPrinciple....................8 1.2.2.2TypesofMFCs.........................9 1.2.2.3DesignConsiderations.....................11 1.2.2.4ChoiceofElectrodeMaterials.................12 1.2.2.4.1Anodematerials:...................12 1.2.2.4.2Cathodematerials:..................13 1.2.2.5Bacteria.............................14 1.2.2.5.1Importantfactorsthattbacterialgrowth:...15 1.2.2.5.2Energycapturemechanisminbacteria:.......15 1.2.2.5.3Electrontransfermechanism:............18 1.2.2.6Substrate............................19 1.2.3MFCModeling..............................20 1.2.3.1AnaerobicDigestion......................20 1.2.3.2MicrobialKinetics.......................21 1.2.3.3Modeling........................22 1.2.3.4MFCModelComparisons...................23 1.2.3.5AnaerobicDigestionModelAnalysis.............23 1.2.4ControlofMFCs.............................25 1.3Conclusion.....................................26 1.4Objectives.....................................27 Chapter2DesignandDevelopmentofanMFC ................ 28 2.1MFCDesignandFabrication..........................28 2.1.1Anode...................................28 2.1.2Cathode..................................30 2.1.3TheFirstPrototype...........................31 2.1.4TheSecondPrototype..........................34 2.1.5OtherSystemComponents........................35 2.1.5.1pHBuPreparationandCalibration............37 2.2InoculumandMediumComposition.......................38 vi 2.2.1Bacteria..................................38 2.2.2Substrate.................................39 2.3DataAcquisition.................................40 2.3.1HPLC(HighPerformanceLiquidChromatography..........41 Chapter3MicrobialFuelCellModeling ..................... 44 3.1MicrobialKinetics................................44 3.2GrowthYield...................................46 3.3PotentialofanMFC...............................48 3.3.1ThermodynamicAnalysis........................48 3.3.1.1Activity.............................49 3.4VoltageOutput..................................51 3.4.1ActivationLosses.............................51 3.4.2OhmicLosses...............................52 3.4.3ConcentrationLosses...........................53 3.4.4TheVoltageOutput...........................54 3.5AControl-orientedModelforMFCs.......................54 3.5.1MassBalances...............................55 3.5.2SummaryoftheControl-orientedModel................57 3.5.2.1StateEquations.........................57 3.5.2.2Output.............................58 3.6ExperimentalModelIden........................59 3.6.1OCPMeasurement............................59 3.6.2HPLCMeasurement...........................62 3.6.3pHmeasurement.............................64 3.7ModelValidation.................................64 Chapter4MFCModelAnalysis .......................... 68 4.1EquilibriaoftheSystem.............................68 4.2StabilityoftheEquilibria............................71 Chapter5ConclusionandFutureWork ..................... 78 5.1Conclusion.....................................78 5.2FutureWork....................................79 BIBLIOGRAPHY ................................... 80 vii LISTOFTABLES Table1.1ComparisonofcharacteristicsincludedinallavailableMFCmodels [39].....................................24 Table2.1Materialsusedforthecathodefabricationandtheirsp.31 Table2.2Materialsusedforthecathodefabricationandtheirsp.32 Table2.3Theingredientsandamountaddedfor3mMacetatein1300mLtotal volumeofmedium............................39 Table2.4Theingredientsandamountaddedfor1mMacetatein2000mLtotal volumeofmedium............................40 Table3.1ParametersidenfortheMFCmodel................63 viii LISTOFFIGURES Figure1.1Typicalfuelcellcomponentsandoperation[6].............5 Figure1.2Anexampleofthepolarizationcurveofafuelcell[7].........6 Figure1.3Typicalschematicofsinglechamberair-cathodemicrobialfuelcell withPEM[9]...............................8 Figure1.4Schematicoftheelectrontransportchain(ETC)throughthebacteria totheelectronacceptor[26].......................18 Figure1.5SchematicofthreeEET(extracellularelectrontransfer)mechanisms usedbyARB(Anode-respiringbacteria)(thatishowCesarI.Torres thebacteriainMFC):a)directelectrontransfer,b)anelectron shuttle,andc)asolidconductivematrix[37]..............19 Figure2.1TheMFCexperimentalsetupusedinthisstudy............29 Figure2.2AnodebrushthatwaspreparedinSML-MSU.............30 Figure2.3CathodematerialthatwasfabricatedinSML-MSU.a)ThePt/C catalystside.b)ThePTFEside...............33 Figure2.4Asingle-chamberair-cathodeMFCthatwasbuiltintheSmartMi- crosystemsLab..............................33 Figure2.5a)ThecomponentsoftheMFCcompartmentforcontinuousorfed- batchsystem.b)Theassembledviewofthecomponents.......34 Figure2.6TheassembledMFCprototypewiththeanode,cathodeandacetate asthesubstrate..............................35 Figure2.7ExperimentalsetupforMFCcharacterizationandmodeldevelopment.37 Figure2.8Thecircuitusedformeasuringthecurrent...............41 Figure2.9ThepHprobewithBNCconnector...................42 Figure2.10dSpaceCP1104..............................42 Figure3.1AnodeopencircuitpotentialversusAg/AgClreferenceelectrodeafter thebacteriainoculation.........................60 ix Figure3.2MaximumpointofanodeOCPversusAg/AgClreferenceelectrode. TheOCPofanodewasdecreasingafterthatpoint..........61 Figure3.3Thesimulationresultsfortheacetateconcentrationchangewithzero input(batchmode)............................62 Figure3.4ThepHvaluestakentwiceaday.....................64 Figure3.5SimulationresultsfortheanodeopencircuitpotentialversusAg/AgCl referenceelectrodeafterthebacteriainoculation............65 Figure3.6AnodeopencircuitpotentialversusAg/AgClreferenceelectrodeafter thebacteriainoculation.........................66 Figure3.7Thesimulationresultsforthebacteriaconcentrationchangewithzero input(batchmode)............................67 Figure4.1Thephaseportraitfortheequilibriumsubspacewithzeroinput(fed- batchmode)................................70 Figure4.2Theeigenvaluesfortuvaluestoshowthebifurcationpoint. Theorangelineshowsthetrajectoryoftheeigenvalueandthe purplelineshowsthetrajectoryofthesecondeigenvalue.......73 Figure4.3Thephaseportraitforthesecondequilibriumpointwithnon-zero inputvalue,0.2..............................75 Figure4.4Thephaseportraitforthesecondequilibriumpointwithnon-zero inputvalue,0.3..............................76 Figure4.5Substrateandbiomassconcentrationstabilizationwiththechangeof thedilutionrate..............................77 x Chapter1 Introduction 1.1Background Energydemandhasalwaysbeenshapingthefutureoftechnologicalopportunities,economy, sociallifeandmanyotherimportantareas.Itisknownthatfossilfuelsarenon-renewable energyresourcesandcontributetovariousenvironmentalissues[27].MFCs(microbialfuel cells)areoneofthepotentialalternativesformeetingtheenergydemandandtheyare environmentallyfriendlyinnature[16],[18],[39],[35].MFCsaredevicesthatusebacteria asthecatalyststooxidizeorganicmatterandgeneratecurrent[4].Bacteriaproduceelectrons bybreakingdownorganicmatterandtransferringtheelectronstotheanode(thesecond electronacceptorafterthebacteria),andthentothecathodewhentheexternalcircuitis completed.Oxygenmolecules(theterminalelectronacceptor)accepttheelectrons,which wfromtheanodetothecathode.However,MFCsarestilllargelyattheresearchlevel,and therearechallengesandbottleneckstoovercomebeforetheybecomecommerciallyviable andwidelyadopted. Therateatwhichbacteriacanoxidizeasubstrate,andtransferelectronsfromthesub- stratetothesurfaceoftheanodehasateonpowergeneration.Whilehaving morebacteriatypicallymeansahigheroxidationrate,theconcentrationofbacteriacould reachsaturationbeyondacertainlevel.Themicroorganismandsubstrateconcentrations arethemainelementsinthedynamicsofpowergenerationinMFCs. 1 Oncebacteriacovertheelectrodesurface,theyformaItisknownthatas thegrowsthicker,masstransfertothebecomesmorelimiting.Therefore, currentdensityislimitedbytheoradvectionofsubstratetotheThere arealsolimitsoncurrentandpowerdensityimposedbytheinternalresistance,whichis relatedtothemembrane(whenpresent)andthematerialsusedforanodeandcathode.The voltagegeneratedbyanMFCismorecomplicatedtounderstandorpredictthanthatofa chemicalfuelcell[18].InanMFC,ittakestimeforthebacteriatocolonizetheelectrodeand manufactureenzymesorstructuresneededtotransferelectronsoutsidethemicroorganism outermembrane.Furthermore,inamixedculture,tbacteriacangrow,resultingin tpotentials.Thepotentialcannotbepredictedevenforpureculturewherethere isonlyonetypeofmicroorganism,duetotheyinkeepingthesystem100%pure (theremightbeothertypesofmicroorganismsthatcaneasilygrowevenwheneverything inthesystemhasbeensterilized).However,therearelimitstothemaximumvoltagesthat canbegeneratedbasedonthermodynamicrelationshipsfortheelectrondonor(substrates) andelectronacceptor(oxidizers). Tobeabletounderstandthephenomenabehindmicrobialfuelcells,itisnecessaryto modelthebehaviourofbacterialsubstrateconsumption.Despitetheirimportance,fewMFC modelsarepresentintheliterature.Researchersdevelopedmodelsbasedonthemicrobial kineticssuchasMonodkinetics,Haldanekinetics(inthepresenceofinhibitors)[28],and triedtounderstandthedynamicsbehindthebacterialgrowthandsubstrateconcentration change,tocontrolachemostatbasedonthebacterialgrowth[30],[31],[32].ForMFCs, someresearcherstakeinaccountthemodels,whicharemoredetailedand providemoreaccurateresultsforcasesinvolvingmicroorganismsforcasesthatdonotneed mediatorstogivetheirelectrons.RittmannandMcCaryexplainedthisapproachin[1]with 2 abroadspectrum.Ontheotherhand,therearestudieswhereresearchersuseasuspended bacteriamodel[18],[39].Bothapproachesacceptablecomparisonswithexperimental results,butoverallthefullunderstandingofMFCsystemsisstilllacking.Theprocessesof electrontransferhavebeenmodeledbefore,butpriorworkonlyconsideredone-dimensional, multi-speciesmodelfortheintheMFC[35].Picioreanu etal. [47]developeda detailed3-dimensionalmodelfortheanodiccompartmentTheprincipalaimofthis modelwastoanalyzeformationandspeciesdistributionwithintheThis wasalsothemodeltotakeintoaccounttmicrobialpopulationscompetingfor spaceandsubstrate.TherehavebeenstudiesoncontrollingMFCsbasedonthe microbialdynamics[45],[38],butexperimentalresultshavenotbeenpresentedtosupport thetheoreticalornumericalresults. MostMFCresearchfocusesonmaximizingpowerproductionbyimprovingtheelectrode materials[55],[65],stackingoftheMFCs[50],usingtbacteriatypes[69],andex- ploringnewdesigns[18].ThereisnotmuchstudyonthecontrolaspectofMFCs,despiteits ontheperformanceoftheMFC.Inthisresearch,nonlineardynamicsofanMFC isstudied.AmathematicalmodelfortheMFCisconstructedandvexperimentally. TobeabletobetterunderstandtheMFCsystemandtoacquiredatafromtheexperiments, asingle-chamberMFCsystemisdesignedandprototyped.Wefurthercharacterizetheequi- libriaofthenonlineardynamicswhenthesubstrateisfedintothesystemcontinuouslyat aconstantrate,andinvestigatethestabilityoftheseequilibriawithJacobiananalysisand phaseportrait. 3 1.2LiteratureReview 1.2.1FuelCells Unlikecombustionengines,fuelcellsproducepowerwithminimalpollutants,whichisone oftheimportantattractivefeaturesoffuelcells.Fuelcellsproduceenergyfromchemical reactionsincontrasttocombustion.However,unlikebatteries,thereductantandtheoxidant infuelcellsmustbecontinuouslyreplenishedtoallowcontinuousoperation[6]. Theprimarycomponentsofafuelcellareanion-conductingelectrolyte,acathode,and ananode,asshownschematicallyinFig.1.1.Fuelcellsaretypicallysingle-chamberedand thischambercontainsbothanodeandcathodecomponents.Inatypicalfuelcell,hydrogen servesasthefueltotheanodeandanoxidant,usuallyoxygen,issuppliedtothecathode compartment.Hydrogenmoleculesarepushedintotheanodewhereachemicalreaction breaksthemdownintoelectronsandhydrogenions.Hydrogenionsarepassedthroughthe electrolyteandreachthecathode.Onthecathodeside,oxygenmoleculesaresuppliedand meettheelectronsandprotonscomingfromtheanode,toformH 2 Oasthebyproduct.There isanoverallelectromotiveforceforhydrogenandoxygentoformwaterasthebyproduct. Thenetreactioninafuelcellissimilartothecombustionofhydrogengas,whichreleases thesameamountofenergy,exceptthatelectricalenergyhasbeenharvestedinsteadofheat. AnodeReaction: 2 H 2 ! 4 H + +4 e CathodeReaction: 4 H + +4 e + O 2 ! 2 H 2 O OverallReaction: 2 H 2 + O 2 ! 2 H 2 O overall = 474 : 4kJmol 1 [8]. overall istheGibbsfreeenergyencebetweenreactantsandproducts.Gibbsfree energyistheenergyassociatedwithachemicalreactionthatcanbeconvertedtowork.The 4 Figure1.1Typicalfuelcellcomponentsandoperation[6]. performanceofafuelcellcanbemeasuredbyobtainingthepolarizationcurve(operating curve).Apolarizationcurveplotsthevoltageofthefuelcellagainstitscurrentdensity.It canbeobtainedbyhavinganadjustableexternalresistorandchangingitsvalue.Today,the polarizationcurveofafuelcellcanbeobtainedwithapotentiostateasily.Itshouldbedone afterthesystemreachesthesteady-stateopencircuitpotential(OCP,whichisthepotential whenthereisnocurrentpassingthroughtheexternalcircuit).Afterthesystemreachesthe steady-stateOCP,thefuelcellcircuitisclosedwithanexternalresistor.Bychangingthe resistorvalueslowly(waitingsometimeateachresistorvalue),onecanmeasurethevoltage acrosstheterminalsandplotthevoltage-currentcurve. Theoperatingcurveandthepowercurvetogethergivevastamountofinformationabout thefuelcell,suchasinternalresistanceofthesystem,optimalexternalresistancetobeused, operatingvoltage,maximumpower,andoverpotentialsofthesystem[18].InFig.1.2,the 5 Figure1.2Anexampleofthepolarizationcurveofafuelcell[7]. slopeofthepolarizationcurveinthelinearpartprovidesanideaoftheinternalresistance ofthefuelcell[7].ItcanbeseeninFig.1.2thattherearethreemainregionsforthevoltage dropwiththechangeofthecurrentdensity.ThetheoreticalOCPcanbecalculatedfromthe Nernstequation[35],[18].However,themeasuredOCPwillbelowerthanthistheoretical valueduetocross-over.Inapracticalfuelcell,somefuelwillfromtheanodethrough theelectrolytetothecathodewhereastheitshouldbeonlyionswhichThiswill reactdirectlywiththeoxygenatthecathode,producingnocurrentfromthecell.Thissmall amountofwastedfuelthatmigratesthroughtheelectrolyteisknownasfuelcrossover[12]. ThemeasuredOCPisalwayslowerthanthetheoreticalvaluedueto, 1-Activationlosses(duetoactivationenergiesandelectrochemicalreactionshappening inthefuelcell) 2-Ohmiclosses(duetoresistanceofthewofionsintheelectrolyteandelectrode) 3-Concentrationlosses(duetomasstransferlimitations) 6 Asthecurrentincreases,thepotentialstartsdecreasingduetotheactivationlosseswhich appearatlowcurrentvalues.Activationoverpotentialsareduetoenergylostforinitiating chemicalreactionsandtheenergylostduetotheelectronstravelingtotheterminalelectron acceptor.Thelinearpartoftheoperatingcurveisduetotheohmiclosses(eventhough therearealsosomenegligibleactivationlossesoccurringinthispart).Ohmiclossescanbe themostimportantpartforanoptimumfuelcelldesign.Theselossesareusuallydueto theinternalconnections,theyofthemembrane,theresistanceofionconduction, etc.Finally,thevoltagedecreasesbecauseoftheconcentrationlosseswiththehighcurrent density.Theselossesarealsocalledmasstransferlossesandareduetoeitherlimitationof theconcentrationofreactantsoroxidantsinthefuelcell. 1.2.2MicrobialFuelCells Microbialfuelcellshavesimilarcharacteristicsasothertypesoffuelcellsexceptthatthey usethebacteriatoreducethesubstrate.However,comparingtomostoftheotherfuelcells, MFCshaveadditionalbWhilegivingusefulelectricity,bacteriaalsotreatwastewater (ifusedasthesubstrate)bybreakingdowntheorganicmatterinwastewater.Mostfuel cellsrelyonexpensivecatalystmaterialswhereasthatoccursnaturallybymicroorganisms inMFCs.Hydrogenfuelcellsrequirecomplexcontrolsystemssincetheyneedtousehighly regulatedstorageanddistributionsystems.MFCscanuseawiderangeoforganicmatter asthesubstrateandcanutilizemicroorganismsthatarecommonlyusedandcaneasilybe foundinnaturalenvironments.MFCscanalsobeoperatedatroomtemperatureunlikethe mostcommonfuelcells,whichrequirehightemperatureandexpensivecontrolsystemsfor that.OneMFCunit(eithersingleordouble-chambered)mighthavealowvoltageoutput (lessthan0.3V),soMFCscanbestackedtoincreasethetotalvoltageoutput. 7 1.2.2.1MFCWorkingPrinciple AscanbeseeninFig.1.3,therearetwohalf-chemicalreactionsoccurringinanMFC,on theanodeandcathodesides,respectively.Theinoculatedbacteriaarekeptintheanaerobic environmentwiththesubstrate,andtheyneedtoattachtoanelectrodetogiveupthe electronsfromthesubstrateconsumptionbecauseofthelackofoxygenasthedirectelectron acceptor.Theelectronchargesintheanodecreatethepotentialbetweenthe cathodeandtheanode.Oncethecircuitiscompletedwitharesistor,theelectronsare transferredtocathodeandreducetheoxygenfromtheairwhichistheterminalelectron acceptor.Oxygenreactswiththehydrogenions(protons)comingfromthechemicalreaction intheanode,whichformsH 2 Oasaresultatthecathodeside.Besides,MFCscanalsobe usedforhydrogenproduction,withaslightlychangeinthedesign. Figure1.3Typicalschematicofsinglechamberair-cathodemicrobialfuelcellwithPEM[9]. InFig.1.3,arepresentationofatypicalsingle-chamberMFCwithaPEM,withacetate asthesubstrateand G.sulfurreducens asthebacteriaisshown.However,membrane-less MFCsaremorewidelyusednowadaysduetothehighcostofPEMs[18]. 8 Inthisstudy,asingle-chamberair-cathodeMFCisusedwith G.sulfurreducens asthe bacteriaandwithacetateasthesubstrate. 1.2.2.2TypesofMFCs MFCscanbedistinguishedbytcharacteristics,suchas,theiroperationconditions, designs,thematerialsused,andthebacteriatypes[42],[21],[18],[25].MFCsaremainly usedinthreemodes:thecontinuousmode,thefed-batchmode,andthebatchmode.In thecontinuousmode,thelimitingsubstratesareconstantlyaddedtothereactor,whilethe outputstreamissimultaneouslyremovedatthesamerate,soastokeepthereactorvolume constant.Designofcontrolalgorithmsistlysimplerforthismodethanthatfor thefed-batchmode,sinceitiseasiertostabilizetheprocessatanequilibriumpoint.In thefed-batchmodethereactoriswithalargeamountofthelimitingsubstrateanda smallamountoftheseedbiomass.Thesystemreachesasteadystateandisharvested[28]. Inboththecontinuousandfed-batchmodes,MFCsactlikeotherfuelcells,whereasifitis abatchsystem,anMFCwouldactlikeabio-battery(inthesenseofabatterywhichuses microorganismstodrivethechemicalreactions).Inthebatchmode,unlikethefed-batch andcontinuousmodes,theMFCcanbeperceivedasaclosedchemostatwhichdoesnothave ordischargingofsubstrate.Thebatchmodeisusedonlyforcollectingthedatain thisstudy. ThereareexperimentalstudiesonthecontinuouswMFCs.Insomestudies[70],[40], theMFCperformanceisobservedbychangingtheHRT(HydraulicRetentionTime).HRT givesaclueofhowlongittakesthesubstratetobeconsumedinareactor.Itisusually expressedinhoursandismathematicallydescribedasthevolumeofthereactordivided bythetwrate.Thedilutionrate(whichispreferredtobeusedinthisstudy 9 insteadofHRT),ontheotherhand,isthetwratedividedbythevolumeofthe reactor.Du( etal. )observedthatintherangebetween10and100mL/min,thewrate isnotthemainfactorsigntlyrestrainingtheMFCperformance[41].Thelattermight beduetotheinhibitors,suchaspHbeinglowerthanthevalue7duetothehydrogen ionsbeingaccumulatedintheanode,orthecompetitionwithothermicroorganismswhich mightdominatethewrateNonetheless,wratechangeinawiderrangecouldbe eondeterminingtheMFCperformance. COD(chemicaloxygendemand)isameasureoforganiccompoundsinwastewater.Itis animportantparameterinMFCsifthepurposeiswastewatertreatment.Inthecontinuous wmode,theCODremovalcouldbeincreasedbyincreasingtheHRT,butthiswouldalso reduceoverallpowerandcurrentgenerationduetoloweraveragesubstrateconcentrations [70].TheseresultsdemonstratethattheHRTwillneedtobeselectedonthebasisofeither optimizingenergyproductionorCODremoval[40]. MFCscanbeoperatedwithaqueouscathodesoraircathodes.TheprincipleofMFC withanaqueouscathodeisthatwaterisbubbledwithairtoprovidedissolvedoxygento electrode,whereasifitisanaircathode,onesideofthecathodeisexposedtotheairwhich allowstheoxygentogetthroughthecathode. MFCscanalsobebytheirdesigns,suchasthesingle-chamber(usuallymembrane- less)rationandthedouble-chamberAsingle-chamberMFChasboth theanodeandthecathodewithinthesamespacewherethechemicalreactionsoccur.A two-chamberMFCrequiresamembranebetweenthecompartmentstolettheions fromonetotheother. 10 1.2.2.3DesignConsiderations AllthematerialsinanMFCsystemshouldbesterilizable(autoclavable)ifonlyacertain typeofmicroorganismsaredesiredintheMFC.Theyinachievinganoptimaldesign comesfromtheconsiderationsofanumberofchallenges,includingchoosing,autoclavable components,keepingtheanodechamberanaerobic,andpreventingtheleakage,etc. PracticalapplicationsofMFCswillrequirethatwedevelopadesignthatwillproduce highpowerandCoulombicInaddition,theeconomicalaspectofcommercial- ization(themanufacturingprocessbeingpracticaltoimplementonalargescale)andthe studiesonmakingmaterialsshouldbeconsidered[18]. ToaddressthelowvoltageoftheMFCs,researchersfocusedonscalinguptheMFCs, andforthispurpose,single-chamberMFCisusedduetoitssimplicityinscalingup.The studyresultsdemonstratethatthespsurfaceareaofthecathodeisthemostcritical factorforscalingupMFCstoobtainhighpowerdensities[23]. Thespacingbetweentheelectrodesishighlythepowerdensity.Twopossible areaseparatorelectrodeassemblyorcloselyspacedelectrodesthatlacka separator,andtheresultssuggestthatseparatorelectrodeassemblydesignscanmore tivelycaptureenergyfromwastewater,butcloselyspacedelectrodeswillbe superiorintermsoftreatmentduetoagreatlyreducedtimeneededfortreatment. Reducingthedistancebetweentheelectrodesusingtheseparatorelectrodeassembly designimprovedperformanceintermsofpowerproduction(0.328Wm 2 )andenergyre- covery(25-78Whm 3 )comparedtothecloselyspacedelectrodesdesign(0.282Wm 2 ,2-34 Whm 3 )[22].Inaseparatorelectrodeassemblydesign,theelectrodesarecloselyspaced butseparatedwithaseparator(topreventshortcircuiting),whereasinacloselyspacedelec- 11 trodesdesign,whereaseparatorisabsent,theelectrodesarespaced2cmfromthecenter pointoftheanodebrushestothecathodePt/Ccatalystsurface[22]. 1.2.2.4ChoiceofElectrodeMaterials TheCoulombice,isastheratiooftotalchargesactuallytransferredtothe anodefromthesubstrate,tomaximumpossiblechargesifallsubstrateremovalproduced current[4].AmainchallengeinconstructinganMFCistoidentifymaterialsandthe architecturethatmaximizepowergenerationandCoulombic.Anotherchallenge istominimizecostandalsotocreatedesignsolutionsthatareinherentlyscalable(scaling updependingonthepurposeoftheuse)[18].Similartootherfuelcells,MFCshavetwo mainelectrodeparts,ananode,acathode,andinsomecases,aseparatingmembrane.The materialsforthesecomponentsarestillasubjectofactiveresearch.Inparticular,itisof interesttoincreasetheandreducethecostfortheelectrodes. 1.2.2.4.1Anodematerials: Electrodematerialsneedtobeinvestigatedforgoodper- formanceinMFCs.Theelectrodematerialsshouldhavecertainproperties,suchashigh conductivity,highporosity,highcatalyticactivitywithoxygen,beingcorrosion-resistant, highsurfacearea,andbeinginexpensive.Fortheanode,thefollowingmaterialsareof- tenusedfor/onanodematerialsduetotheirstability,highelectricconductivity,andlarge surfacearea[18]: carbonpaper,cloth,foams,andRVC(reticulatedvitreouscarbon); graphiterods,felts,foams,plates,andsheets; graphitegranules; 12 graphiteersandbrushes; conductivepolymers; metalsandmetalcoatings. Amongthose,carboncloth,carbonfelt,graphitefelt,carbonmeshandgraphiteerbrushes aremostcommonlyusedinMFCs.Themainreasonwhythegraphiteerbrushesare frequentlyusedisthattheyhavethehighestspsurfaceareaandporosity[18]. 1.2.2.4.2Cathodematerials: Thecathodesideismorecomplicatedintermsofma- terialscience.Thechemicalreactionthatoccursatthecathodeistoengineeras theelectrons,protonsandoxygenmustallmeetatacatalystinatri-phasereaction(solid catalyst,air,andwater)[18]. Thechoiceofthecatalystiscrucialinthecathodematerialsinceitthesivity ofoxygenintheMFCdirectly.Althoughplatinum-coatedelectrodesaremoretand superiorthanotherelectrodesinpowerproductionduetohighercatalyticactivitywith oxygen,theyarenote[19]. Thematerialsoftenusedforthecathodeare[18] carboncathodeswithPtcatalysts; carboncathodeswithnon-Ptcatalysts; plaincarboncathodes; tubularcarbon-coatedcathodes; aqueouscatholytes; 13 PtandPt-coatedmetals; metalsotherthanPt; biocathodes. BecausethePTFE(polytetraethylene)layerscoatedontheair-cathode letstheoxygengetthrougheasily,whichdoesnotrequireexternalairspargingtothecathode, air-cathodeswithPtcatalystaremostcommonlyusedinMFCs.Manyresearchershave chosentouseair-cathodes,asthesetypesofelectrodeswillultimatelybethetypeofcathodes usedinlargersystems[18].Inthisstudy,air-cathodewihPtcatalystisused;however, somerecentstudieshavefocusedonmesoporousnitrogen-richcarbonmaterialsascathode catalystsasanalternativetoPtcatalyst[20]. 1.2.2.5Bacteria MFCsstartedwiththediscoveryof E.Coli bacteriaproducingelectricity.M.Pottermade theattempttoproduceelectricityfromthismicroorganismwithoutamediator(a chemical,suchasneutralred,thattransferselectronsfromthebacteriaintheMFCtothe anode),whereheusedaplatinumelectrode[25].However,since E.Coli couldnottransfer itselectronswithoutamediator,thisstudydidnotcatchmuchinterest.In1980s,itwas discoveredthatthecurrentdensityandpoweroutputcouldbegreatlyenhancedbythe additionofelectronmediators[42]. RecentstudiesshowthatthereareawiderrangeofbacteriaoptionstouseinMFCs, andthereisnoneedformediatorssincemostofthebacteriacanusespecialmethodsto givetheirelectronstotheelectronacceptor(anode).Thesebacteriathatcantransferthe electronsoutsideoftheircellsarecalled exoelectrogens [18].Mostfrequentlyusedbacteria 14 forMFCswiththispropertyareShewanella,RhodoferaxandGeobacterstrains. 1.2.2.5.1Importantfactorsthatbacterialgrowth: Chemicalreactionsare bythetemperature,sobacterialgrowthisalsobythetemperaturechange intheenvironment.Foreachtypeofbacteria,thegrowthrateincreaseswiththetemper- ature,and,ingeneral,theratedoubles[1].Eventhoughtemperatureisnotabigissue forMFCsingeneral,temperaturesabovethenormalrangeforthespeciescandestroythe enzymesandtheorganismmaydie. ThepHalsoectsthegrowthandmostspeciesofbacteriahaveanarrowpHrange forgrowth,andformostorganismsthisrangeliesbetween6and8[1].Thestudyalso suggeststhat,thedesignandoperationofanMFC-basedtreatmentsystemmustconsider theoptimumpHconditionsrequiredforgrowthofthebacteriaofinterest. Ifthebacteriarequireananaerobicenvironment(forexample,Geobacters),thepresence ofmolecularoxygenwouldtthegrowthabilityofbacteriatly.Fortheterminal electronacceptor,somebacteriacanusenitriteorsulfateinsteadofoxygen.Butsinceoxygen iseasilyfoundintheairandthereductionofoxygenwouldhavemoreGibbsfreeenergy,it isgenerallypreferredtouseoxygenastheelectronacceptorinMFCs. Microbestransferelectronstotheelectrodethroughanelectrontransportsystemthat eitherconsistsofaseriesofcomponentsinthebacterialextracellularmatrixortogetherwith electronshuttlesdissolvedinthebulksolution[42].Moststudiesthusfarhavefocusedon investigatingtheelectrontransfermechanismsthatenableGeobactertoreducethe electrode[10]. 1.2.2.5.2Energycapturemechanisminbacteria: Theelectroncarrierscanbedi- videdintotwotclasses,thosethatarefreelythroughoutthecell'scyto- 15 plasmandthosethatareattachedtoenzymesinthecytoplasmicmembrane[1].There areco-enzymesinthebacteriawhichcarrytheusefulenergy.Theco-enzyme NADP + is involvedinanabolicreactionswhile NAD + incatabolicreactions.Theparticularelectron carriersthatoperateinagivencelldependupontherelativeenergylevelsoftheprimary electrondonorandtheterminalelectronacceptor[1]. NAD + extractstwoprotonsandtwoelectronsfromamoleculebeingoxidizedandin turnisconvertedtoitsreducedform, NADH .Thereactionsof NAD + is, NAD + +2 H + +2 e = NADH + H + (1.1) TheGibbsfreeenergyforthisreactionis G =62kJ(1.2) Similarly, NADP + takestwoprotonsandtwoelectronsfromthesubstrateandform NADPH : NADP + +2 H + +2 e = NADPH + H + where G =62kJ(1.3) WhentheGibbsfreeenergyispositive,itmeansthatenergymustbetakenfromthe organicmoleculeinorderfor NADH tobeformed.Whenthe NADH inturngivesupthe electronstoanothercarrierandisreducedbackto NAD + (Fig.1.4),italsogivesupthe chemicalenergy,whichmaybeconvertedtootherusefulforms[1]andformthepartof ETC(electrontransportchain). Innaturalenvironmentsoxygenistheterminalelectronacceptorforthebacteria,and theenergyreleasedastheelectronsarepassedthroughachainofelectroncarriers(Fig.1.4) 16 tooxygencanbedeterminedfromtheoverallGibbsfreeenergychangeofthe NADH and O 2 halfreactions, NADH + H + = NAD + +2 H + +2 e G = 62kJ(1.4) 1 2 O 2 +2 H + +2 e = H 2 O G = 157kJ(1.5) Netreactionis: NADH + 1 2 O 2 +2 H + = NAD + + H 2 O G = 219kJ(1.6) Therefore,theenergytransferredalongwithelectronsfromanorganicchemicalto NADH isreleasedtosubsequentelectroncarriersandultimatelytooxygeninaerobicres- piration.However,inMFCsaportionofthisenergyisgoingtobecapturedbytheanode beforebyoxygen.Thisenergytransferresultsin 219kJpermoleof NADH forusebythe organismintheaerobiccase.Thebacteriaisusingaportionofthisenergyformaintenance, cellsynthesis,growth,etc.Theprimaryexampleofanenergycarrierforthispurposeis adenosinetriphosphate( ATP )(Fig.1.4).Whenenergyisreleasedfromanelectroncarrier, itisusedtoaddaphosphategrouptoadenosinediphosphate( ADP )[1], ADP + H 3 PO 4 = ATP + H 2 O G =32kJ(1.7) Geobacterbelongstodissimilatorymetalreducingmicroorganisms,whichproducebio- logicallyusefulenergyintheformofATPduringthedissimilatoryreductionofmetaloxides underanaerobicconditionsinsoilsandsediments[42]. 17 Therefore, NAD + capturesprotonsandforms NADH + .Then NADH through bacteriacytoplasmandforms NAD + andoneprotonbyreleasingelectrons(energy).Some ofthisenergyisusedbybacteriatoform ATP whichtheyneedforcellsynthesisandfor cellmaintenance.Wheneverbacterianeedsenergy,itreduces ATP to ADP andthis ADP latertakesthesamecycletoform ATP from NADH to NAD + reaction. Figure1.4Schematicoftheelectrontransportchain(ETC)throughthebacteriatothe electronacceptor[26]. 1.2.2.5.3Electrontransfermechanism: Afterbacteriacapturetheenergyfromthe substrate,theygiveouttheelectronstoanelectronacceptorwhichistheanodeinMFCs. Itisstillnotfullyunderstoodhowbacteriatransfertheirelectronsoutsideoftheirouter membranes.However,thereisanagreementonthreettypesofextracellularelectron transfer(EET),bytmicroorganisms.InFig.1.5,thethreediscoveredEETare depicted. ThemechanismisexplainedasthatbacterialelectroncarriersthroughtheETCgive theelectronsthroughtheoutermembranetoasolidelectronacceptor.Bacteriausingthis 18 mechanismrequiredirectcontactwiththesolidelectronacceptorand,thus,cannotforma [37].Thesecondmechanismneedsamediator,whichhelpscarrytheelectronsfrom thebacterialoutermembranetothesolidelectronacceptor. Thethirdproposedmechanisminvolvesasolidcomponentthatispartoftheextracellular matrixandisconductiveforelectrontransferfromthebacteriatothesolidsurface [37].Thismechanismissupportedbytherecentdiscoveryofthepossibleroleofcellular piliasnanowires(whichsomebacteria,suchas G.sulfurreducens ,producetoattachthe electrode)[17],whicharebeingcharacterizedfortheircapabilitytoconductelectrons[37]. Figure1.5SchematicofthreeEET(extracellularelectrontransfer)mechanismsusedby ARB(Anode-respiringbacteria)(thatishowCesarI.TorresdethebacteriainMFC): a)directelectrontransfer,b)anelectronshuttle,andc)asolidconductivematrix[37]. 1.2.2.6Substrate SubstrateisconsideredtobeakeyfactorforMFCssinceithasthesourcefortheorganic matterwhichbacteriausetoextractenergy.Therearealargenumberofsubstratetypes usedinMFCs. 19 ItistocomparetheperformanceofMFCsfromtheliteraturebythesubstrate used,duetothetoperationalconditionsineachstudy.Therearesomemeasures, suchascurrentdensity,thatcanhelpusunderstandtheofsubstrates.Theunitused forcurrentdensityisusually(mA/cm 2 ).ThemajorityofMFCresearchersuseacetateas thesubstratesinceitshowthehighestenergyoutputfromMFCscomparingtoothersingle substratetypes[21]. Therearestudiesthatpointtotheimportanceofthesubstrateonthepowerdensity andpresentrelevantmathematicalmodels[1],[18],[35],[37].Forexample,itisshown[23] thatthesubstrateconcentrationhasatonthepotentialontheanodeside butnotcathodeperformance,whilethesolutionconductivityhasatonthe cathodebutnottheanodeperformance. 1.2.3MFCModeling Inthissection,forthepurposeofmodelingandcontrollingMFC,therelevantliteratureis criticallyreviewed.ToderiveamathematicalmodelforanMFC,oneshouldstartwithun- derstandingthebacteriacharacteristics,microbialkinetics,andanaerobicdigestion.There arevastamountofliteratureaboutthemicrobialkineticswhichexplainsthemicroorganism behaviorindetails[1],[13],[14],[15]. 1.2.3.1AnaerobicDigestion Anaerobicdigestionisaprocesswhereorganicmatterisdegradedintoamixtureofmethane, carbondioxide,andbiomass[51].Amongthediverseprocessdesignsandfor anaerobictreatmentprocesses(e.g.,anaerobicsuspendedgrowth,wanddoww anaerobicattachedgrowth,anaerobiclagoons,wanaerobicsludgeblanket),somepre- 20 viouslydevelopedanaerobicdigestionmodelscangiveimportantinformationaboutkinetics ofreaction,transport,andspacelimitationsfortheMFCanode[39].Theanaerobicdiges- tionmodelsconsideredinthisstudyinvolveanaerobicsuspendedgrowthandthe attachment. 1.2.3.2MicrobialKinetics Therearetwocommondescriptionsforthesubstrateconsumptionkineticsandthemicroor- ganismgrowthrate,MonodkineticsandtheHaldanekinetics.Monodkineticswasnamed aftermicrobiologistJacquesMonod,in1940s,andisalsocalledMichaelis-Mentenkinetics [11].Intheliterature,Haldanekineticsusuallyisusedfortheoptimalcontrolpurposefor theanaerobicdigestion,suchasthatin[59].OneofthereasonsthatHaldanekineticsis usedinnonlinearanalysisisbecauseitconsiderstheinhibitors.Andwiththat fromtheMonodkinetics,aninterestingphenomena,thesingulararc,occursintheoptimal controlanalysis[67].However,inthisstudyMonodkineticsisconsideredduetoitsbetter accuracytorepresentMFCmodels.TheMonodequation, = max S K + S (1.8) expressesthat,atahighsubstrateconcentration,theprocessisatitsmaximumrate,while atalowsubstrateconcentrationthesubstratebecomesratelimitingforthesystem.Here, max isthemaximumspgrowthrate, S istheconcentrationofrate-limitingsubstrate, and K istheconcentrationgivingone-halfthemaximumrate.Eq.(1.8)representsthe microbialkineticsforonesubstrate.Itcanalsobemoiftherearemorethanone 21 substrateinthemedia[35];forexample,fortwosubstrates S 1 and S 2 , = max S 1 K 1 + S 1 S 2 K 2 + S 2 (1.9) Overtheyears,manytanaerobicdigestionmodelswerestudied[51],[56],but theiruseswerelimitedtospapplications[39].Duetothetapproaches,IWA (InternaionalWaterAssociation)developedageneralizedanaerobicdigestionmodel,ADM1 (Anaerobicdigestionmodelno1)[58].TheADM1modelprocessesincludedmanydetails, suchas,acidogenesisfromsugars,acidogenesisfromaminoacids,acetogenesisfromlong chainfattyacids,acetogenesisfrompropionate,acetogenesisfrombutyrateandvalerate, aceticltasticmethanogenesisandhydrogenotrphicmethanogenesis[58].Thecomplexityof thismodelrequireslargecomputationalandcreateschallengesinmodelparameter idenwhichmeansthatparameterestimationalgorithmscouldnotsuchalarge numberofparameterswithinreasonablelevels[53].Forthatreason,ADM1is nottakenasthebasemodelforthisstudy. 1.2.3.3Modeling ADM1didnotconsiderthegrowth;insteaditassumedawell-mixedchemostatforthe anaerobicdigestion.IntheMFCsbacteriausuallyformaontheanodesurface,which becomesasolidconductivelayerinthesenseofelectronaccepting.In[52],theauthorsused severalmicrobialspeciesandobservedthecompetitionforspaceandsubstrate.Theyused thecontinuumapproach,massbalanceequations,andexperimentalobservationstodescribe theofrelativesubstrateconcentrationsonbioperformanceandcomposition[52]. ThemodelingofformationinMFCscanbehighlycomplex.Inthestudy[48],the 22 authorsmodeledamixedpopulationformationinthreedimensions.The ofseveralsubstratesandthegrowthoftmicroorganismsweremodeledusingPDEs (partialtialequations)duetothenatureofthebmodeling.Inthepaper[35], theauthorsdevelopedadynamic,one-dimensional,multi-speciesmodelforthein theanodeofanMFC.Theyconcludedthattheconductivitystronglythe electrondonor,currents,andthebiomassdistribution.Theyusedtheanodepotential toderivethemodel.In[37],theauthorsaimedtoevaluatehowwelleachextracellular electrontransfer(EET)mechanismcanproduceahighcurrentdensitywithoutalarge anodepotentiallossbyusingthemodel.Thecomplexityofthesemodelsissimilar tothatofADM1,whichrequireslargecomputational 1.2.3.4MFCModelComparisons SomeMFCmodelsaretoocomplextosolveandsomeofthemarenotwellsuitedforcontrol. Table ?? summarizesthecharacteristicsofavailableMFCmodelsintermsofanodeor/and cathodetypes,microorganisms,existence,andtheconvergencey(whether themodelswoulddemandlargecomputationaltobesolved). InMFCresearchwiththewastewaterasthesubstrate,multi-speciesareoftenconsidered. Inthisstudy,pureculture(single-microorganism)isconsideredandthenecessityoffast convergenceistakenintoaccountinthemodelanalysis. 1.2.3.5AnaerobicDigestionModelAnalysis Modelingofanaerobicdigestionandtwomainkinetics(substrateutilizationandthemicro- bialgrowthrate)hasbeenwellstudied[2],[57].Thesemodelshavebeeninvestigatedfor theirsystempropertiesinafewcases,includingstabilityanalysisoftheequilibria,feedback 23 Table1.1ComparisonofcharacteristicsincludedinallavailableMFCmodels[39]. Model TypeofMFCmodel Multi-species Easeof convergence ZhangandHalme (1995) Anodeandcathode No No Yes Zengetal: (2010) Anodeandcathode No No Yes Macusetal: (2007) Onlyanode No Yes Yes Hamelersetal: (2011) Onlyanode No Yes Yes Picioreanuetal: (2007) Anodeandcathode Yes Yes No Picioreanuetal: (2010 a ) Anodeandcathode No No No *Thismodelassumedinertandactivebiomasscompetingonlyforspaceontheanode surface. stabilization,robuststabilizationwithLyapunovmethod,andoptimalcontrolofthesystem. AstabilizingfeedbackcontrolwasdesignedforHaldane-Monodmodelofbacterialgrowth [66].Morerecently,theauthorsof[68],studiedoptimalcontrolofanonlinearfed-batchbio- processusingapredictiveapproach.Stabilityanalysis,nonlinearanalysis,andcontrolof anaerobicdigestionwereperformedusingsimilarmodelkinetics[31],[43].Forthestability analysisofthechemostatsappropriateLyapunovfunctionswerechosenbysomeresearchers [63],[28].Onthecontrolside,therehavebeenstudiesfocusingonfeedbackdesign[30]for tpurposes,suchastoregulatetheorganicpollutionlevel[61]usingthesimilaranaer- obicdigestionmodel.Therearealsostudiesonnonlinearadaptivecontrolforbioreactors [62]inwhichtheauthorsproposedanonlinearcontrollerandprovedtheglobalasymptotic stabilityoftheclosed-loopsystem. Despitetheaforementionedprogresses,theworkscitedabovemostlydealwithbioreac- tors,notforMFCs.EventhoughanMFCincludeslms,itsuniquecharacteristics newopportunitiesformodeling,analysis,andcontroldesign. 24 1.2.4ControlofMFCs Therearenumerousstudiesonthecontroloffuelcellssincetheyhavecomplexsystemswith numerousequipmentcomponents,suchaspumps,compressors,storagetanks,wmeters, andsensors.Ontheotherhand,forthecontrolofMFCs,duetoitsrelativelynewnature, researchhasbeenverylimited. Theoryandexperimentswiththenutrientwcontrolformaximizingtheamountof microorganismswereexploredbytheauthorsof[66].Inthisstudy,astabilizingfeedback controlwasdesignedforHaldane-Monodmodelofmicrobialgrowthof E.Coli .Inanother study[61],acontrollawwasproposedtodrivethemodeltoadesiredsetpointforanyinitial operatingconditionsforananaerobicdigestion.TheauthorsconcludedthatCOD(chemical oxygendemand)ofwastewatercanbeestimatedon-linewithouttheneedforsensors.The latterworkwasonlyfocusedonachemostatwhichhadananaerobicdigestion,andthusis tfromanMFC.. ResearchershaveconductedstudiesonparameterestimationfortheMFCs[39].In thisstudy,theauthorsconsideredmulti-speciesformicroorganismsandwastewaterasthe substratefortheanaerobicdigestion.TheymodeledMFCbasedonthemixedsystem(not assumption,exploredtheoftheexternalresistanceontheperformance,and optimizedthesubstrateconsumptionbystagingMFCs.Theyalsoconcludedwith R int and E OCP estimationsasfollows, R int = R MIN +( R MAX R MIN ) exp K R X (1.10) E OCP = E MIN +( E MAX E MIN ) exp ( 1 K R X ) (1.11) 25 R MIN isthelowestobservedinternalresistanceandR MAX isthehighestobservedinternal resistance.E MIN isthelowestobservedOCPandE MAX isthehighestobservedOCP.K R is theconstantwhichdeterminesthecurvesteepnessanditcanbeobtainedusingthevoltage measurementsfromtheoperatedMFC.The R MIN , R MAX , E MIN and E MAX valueswere obtainedfromthepolarizationtests.Inthesamestudy,theyusedmediators,butinmost ofMFCstudiesmediator-lessbacteriaareusedduetotheextraexpenseofmediators[40], [41],[45],[18],[10],[46]. Therehavebeensomeattemptsonmaximizingthepoweroutputwithpowermanagement systemsandPIDcontrollers[45],[38]forMFCs,buttherehasbeenlittleexperimentalwork inthesestudiestosupportthesimulationresults. 1.3Conclusion Inthischapter,anintroductiontoMFCsiscarriedoutwiththebackgroundandanextensive literaturereviewonthemodelingandcontrolissues.Ithasbeenconcludedthatthereisnot adequateworkoncontrolofMFCs,eventhoughitisanimportantareawhichisdirectly relatedtotheMFCperformance,costreduction,andbetterunderstandingofthesystem. Ithasbeenrealizedthattherearetmodelsdescribingmicrobialkinetics,andin thisstudyMonodkineticsisselected,sinceitdoesnotincludetheinhibitorctandis abetterrepresentationforMFCsystems.Forthemodelanalysis,therehasbeenresearch onanaerobicdigestionandmicrobialkineticsmodelanalysisforbioreactors.However,such analysishasnotbeenappliedtoMFCmodelswiththespparametersforMFCdynamics. 26 1.4Objectives Thegoalofthisresearchistounderstandthemathematics,theory,andmechanismbehind theMFCsystemswiththeavailablemodels,andtofurtherinvestigatethedynamicmodel inordertodevelopanMFCcontrolsystemforitsperformanceoptimization. Therefore,theobjectivesofthisstudyare 1.TodesignanMFCwhichcanserveinbatch,fed-batchandcontinuousmodesandbe usedasaplatformforcollectingexperimentaldata. 2.ToamodelforelectricitygenerationfromMFCsthatcanbevalidatedwithex- periments.Substrateandmicroorganismconcentrationsaretobeusedasthestate variablesinthemodel. 3.Toanalyzethesystembasedonnonlinearsystemstheory,includingthestabilityofthe equilibriumpoints. 27 Chapter2 DesignandDevelopmentofanMFC Inthischapter,thedesignandprototypingofanMFCarepresented.ThisMFCissub- sequentlyusedinmodeldevelopmentandanalysis.TheexperimentsetupusedinSMLis showninFig.2.1.ApotentiostatisusedtocollectthedataandsendittothedSpace ControlDesk.D-SpaceusesMatlab/Simulinkinitsinterface.Thesystem(bothMFCand thestoragecontainers)werebubbledwithnitrogenconstantly,fromthenitrogencylinder. ThereareautoclavedsyringeneedlesonthetopofbothMFCandstoragecontainerstolet theoxygenbubbledout.Filtersareusedtokeepthenitrogengassterile.ToshowthepH sampleswerecollectedfromtheMFConaregularbasis.Samplesweretaken withtheaidofasterilesyringeandsamplesarealsousedfortheHPLCanalysisinReguera labatMSU. 2.1MFCDesignandFabrication 2.1.1Anode TheanodematerialusedintheMFCinthisstudywascarboner(PANEX3550K,Zoltek) brushwithtwotwistedTi(Titanium)wires.Thespecforthebrush(TheMillRose Company,Mentor,OH,USA)are1 : 989inchesindiameter(Fig.2.2),2 : 75inchesinlength forthebrushpart,and4inchesinoveralllengthwiththeTipartincluded.Carboner 28 Figure2.1TheMFCexperimentalsetupusedinthisstudy. 29 brushesweresoakedintheacetoneovernightandthenheat-treatedat450 Cfor30min[33] inthePhysicsLaboratoryatMSU. Figure2.2AnodebrushthatwaspreparedinSML-MSU. Thereasonwhyabrush-typeelectrodewaschosenwasthatithashighersurfacearea thanothercarbonfeltanodes.Intheexperiments,onlysomeportionoftheanodebrush wasincontactwiththemediumbecauseoftheshortlengthoftheTiwireontheanode brush(Fig.2.6).Theactiveanodebrushsurfaceareawascalculatedtobeapproximately 0.67m 2 . 2.1.2Cathode ThecathodewasmanufacturedintlaboratoriesatMSU.Thecathodesweremade byapplyingplatinumandfournlayersonacarbonclothasdescribed in[24].ThematerialsusedforthisprocessanddetailedinformationareshowninTable2.2. First,thecarbonbaselayerwaspreparedusingthecarboncloth,carbonblackpowder, and40%PTFEsolution.This40%PTFEsolutionwaspreparedfrom60%PTFEsolution (Table2.2),bydilutionwithDI(deionizedwater).Thewholemixturewaskeptintheplastic samplevialtube.Solidglassbeadswereaddedintothetubetohelpforminghomogeneous mixture.Themixturewasmixedwithavortexer(Vortex-Geniemixer,S8223,Scien 30 Table2.1Materialsusedforthecathodefabricationandtheirspns. Material Vendor Address Sp Carboncloth ElectroChemInc. MA,USA T dimensions 19cmx19cm PTFE (Pethylene) Sigma-Aldrich USA 60wt% dispersion inH 2 O 10%Pt/Ccatalyst ElectroChemInc. MA,USA Weight10wt% platinum, Vulcan XC-72carbon, amount5grams p resinsolution Sigma-Aldrich USA 5wt.%inmixture oflower aliphatic alcohols andwater, contains45%water Carbonblackpowder CabotCorp. GA,USA VulcanXC72 Conductive CarbonBlack Propanol AlfaAesar MA,USA 2-Propanol, Spectrophoto- metricGrade, 99.7+% SolidGlassBeads PropperManufacturing Co.Inc. NY,USA 3mmindiameter 31 Industries,Inc.,USA)intheRoboticsandAutomationLabatMSU.Thenthismixture wascoatedonthecarbonclothwithasmall,softpaintbrush.Thecarbonclothwiththe coatingwasheat-treatedonaheat-resistantglassceramicplate(McMasterCarr)inafurnace (PhysicsLaboratoryatMSU)at370 Cfor25min.Secondly,60%PTFEsolutionwasapplied onthepreviouslycoatedsidetoformthelayer.Theoptimumlayernumberwas foundtobefouraccordingto[24].ThecarbonclothwiththePTFEcoatingwasheat-treated onthesamefurnaceat370 Cfor12minbetweeneachlayer.Finally,thecatalystlayerwas applied.Thematerialsusedforthisstagewere10%Pt/C,propanol,andDI(Table 2.2).Moredetailedinformationaboutthiswholeprocesscanbefoundin[44]. ThePt/Ccatalystsideoftheair-cathodeisincontactwiththemediainMFCwhereas thePTFElayerhelpstheoxygengetthroughtoformtheH 2 O(Fig.2.3). Figure2.3CathodematerialthatwasfabricatedinSML-MSU.a)ThePt/Ccatalystside. b)ThePTFEside. 32 2.1.3TheFirstPrototype Inthisstudy,asingle-chamberrectangularprism-shapedair-cathodeMFC(Fig.2.4)was builtinSML.Toprototypethisdesign,round,impact-resistantpolycarbonatetubeand sheets(McMasterCarr)wereused.Thetotalliquidvolumewas230ml. Figure2.4Asingle-chamberair-cathodeMFCthatwasbuiltintheSmartMicrosystems Lab. Thesystemwouldbeoperatedinabatchmodeandwithtsubstrateconcentra- tions;however,thecathodesectionoftheMFChadtoomuchleakageforitstotalliquid volume.Theleakageproblemcouldnotberesolvedwithafewtattempts,becauseof theyassociatedwiththechoiceofautoclavablematerialsandbecauseoftheanaer- obicrequirement.Forthatreason,analternativebottle-shapeddesignwasconsidered,as discussednext. 33 2.1.4TheSecondPrototype Inthisnewdesign,therewasstillleakageonthecathodepart;however,itcouldbesolved bysealingthecathodeextensionwithwaterproofautoclavablesilicon(ACEHardware,MI, USA).Theleakingproblemwasfromtheexternalpressure,whichcamefromthenitrogen cylinderforkeepingthesystemanaerobic.Inthecurrentdesign(Fig.2.7)thespacing betweentheanodebrushandthecathodewas2inches.ThisMFCsystemcanbeusedfor batch,fed-batch,andcontinuoussystems. Figure2.5a)ThecomponentsoftheMFCcompartmentforcontinuousorfed-batchsystem. b)Theassembledviewofthecomponents. ThecomponentsshowninFig.2.5include,1)cathodethatwasfabricated,2)auto- clavabletubingforthesubstratew,3)thearmextensionforthecathodeplacement,4)a glasspartwhichsandwichesthecathode,5)o-ring,6)clipstoholdthecathodebetweenthe glassandtheextensionarm,7)iwextensionbarbed8)wextensionbarbed 9)bluebutylstopper,and10)theextensionforthereferenceelectrodeandforthe N 2 w. 34 Figure2.6TheassembledMFCprototypewiththeanode,cathodeandacetateasthesub- strate. 2.1.5OtherSystemComponents TheMFCsystemwasdesignedtobeusedforboththebatchmodeandthefed-batchmode (eventhoughitcanbeusedalsoforthecontinuousmode),anditrequirestheabilityto modulatethemediumw.Twosolenoidvalveswereusedwithonenormallyopened(NO) (Honeywell,modelno:71225SN2EF00N0C111B6,MI,USA)andtheothernormallyclosed (NC)(Parker,modelno:71215SN2MF00N0C111P3,MI,USA).Thespforthe valveswere,120 = 60volts/Hz,10Watts,,750psi.Thevalves1 = 4incheswitha3 = 64inches size( C v =.005).TheNCvalvewasusedforwwhiletheNOvalvewasusedfor wtotheMFC. Fordataacquisitionandsystemcontrol,dSpaceControlDesk5.1wasusedwithacom- puter.DSpacerequired 10 = +10voltasaninput/output,soSSRs(SolidStateRelays) (Opto22,CA,USA)wereused,alongwiththefuse,toconvertthedigitalsignal(VDC)to 120VACtooperatethesolenoidvalves. 35 Autoclavableglassmedia/storagebottles(UnitedStatesPlasticCorp.,OH,USA)used tostorethemediaandtooperatetheMFC.Theglasseswerere-shaped(Glassworkatthe ChemistryBuilding,MSU)basedonthepurposeofthisstudy.ThebottlewheretheMFC wasoperatedwas250ml,theonewhichstoredtheinputmediumwas1000mlandthe tstoragebottlewas500ml. Thesystemneedstobekeptanaerobic.Therefore,anitrogencylinderwith99 : 999% purity,304SCF(standardcubicfoot)(MSUStores)keptpumpingthenitrogenintothe MFCbottletogetoxygenoutofthesystem.Tolowerthehighpressureofthenitrogen cylinder,aregulator(SMITHEquipment,SD,USA)wasusedsothattheoutletpressurewas alwayslessthan5psi.Splly,boththeMFCbottleandtheinputstoragebottleneeded tobekeptanaerobicsothenitrogengaswassplittedtothebottleswithasplitter.Tubing usedforthispurposewasnorprenetubing(Cole-ParmerInstrumentCompany, IL,USA).Sterileneedlesandluerlockswereusedtogetthroughthebluebutylstoppers (FisherScienCompany,PA,USA)totheMFC.Aneedlewasattachedtothetopofeach bottletoreleasetheoxygen.PTFEsyringe(FisherScienCompany)wereattached totheneedlestokeepthenitrogengassterile.Autoclavabletubing(McMasterCarr)with brass-barbedwereusedtoletthesubstratewinthesystem. AnAg/AgClreferenceelectrodewithwireconnectors(BASi,IN,USA)wasused fortheanodeelectrodepotentialmeasurement.AnexternalTiwire,0.5mmindiameter, wasusedonthecathodesidefortheclosed-circuitdataacquisition. Therewereothermaterialsandsafetyequipmentusedforthefabricationandoperation, suchas50mLconicalpolypropylenecentrifugetubesformixingthesolutionsandpreparing thepHdistilledwaterforthepHcalibration,acetonefortheanodebrushtreatment, glasses,faceshield,nitrilepurpleglovesandgogglesforsafety(MSUstores).Allcomponents 36 Figure2.7ExperimentalsetupforMFCcharacterizationandmodeldevelopment. oftheMFCsystemwereautoclavedfor30minunder24psipressurevaluewith121 C withgravitycycle(steamdisplacesairinthechamberbygravity,i.e.withoutmechanical assistance,throughadrainport)atthesametimeinanovenattheRegueralabatMSU. 2.1.5.1pHPreparationandCalibration ThepHprobeneededtobecalibratedeachtimebeforeitwasused.Forthecalibration, threetpHsolutions(ofpH4,7and10)wereprepared.capsules(order codePHB)wereusedforpreparingthepHsolutions.Powderfromeachcapsulewas addedinto100mLofdistilledwaterinaplasticbottle.Then3dropsofpreservative (MicroEssentialLaboratory,Broklyn,NY,USA)wereaddedtopreventmoldgrowthand toaddcolortothesolutionsoeachsolutioncanbeiden BluepH10(withtoleranceof0.02)at25 C GreenpH7(withtoleranceof0.02)at25 C OrangepH4(withtoleranceof0.02)at25 C 37 Afterthewereprepared,thepHprobewasusedtohavethreetvalue points.Thesethreepointswereusedtohavethefollowingequation.ForthepHmea- surement,samplesweretakenwithasyringe(sliptip20mL,MSUStores)fromtheMFC container. pHvalue = 2143 ( Voltagevalue ) 33 : 2884(2.1) 2.2InoculumandMediumComposition 2.2.1Bacteria G.sulfurreducens strainPCAwasusedforthisstudy.Itwasinoculatedforthecontrolpur- posewiththefed-batchmodeinourexperimentandwasroutinelyculturedanaerobically intheDBAF(DBwithfumarateandacetate)medium,whichwasDBmedium[3]sup- plementedwith20mMacetateastheelectrondonorand40mMfumarateastheelectron acceptor. Na 2 SeO 4 (1mM)wasalsoaddedtostimulategrowth,asreportedelsewhere[3].The MFCwasinoculatedwithcellsuspensionsasdescribedpreviously[3],exceptthattheelectron donorintheanodechamberwasacetate,whichhadaninitialconcentrationof3mMfor experiments,andforthelatterexperiments,theacetatemediumhadaninitialconcentration of1mM. Thebottlesforbacteriaweretopoutatopticaldensityat OD 600 (opticaldensityat600 nanometer)of0.6,and100mLofthisculturewasspundown,washedtwiceandthenre- suspendedin10mLofmediumfortheinoculationtotheMFC.Theinitialcellconcentration wascalculatedtobeapproximately5x10 10 cells/mL. 38 Table2.2Theingredientsandamountaddedfor3mMacetatein1300mLtotalvolumeof medium. Ingredients Amount added Units Final concentration 10XDBStock 130 mL 100XDBMineralMix 13 mL 100XDLVitaminMix 13 mL 0.75MNaAcetate 5.2 mL 3 KH 2 PO 4 2.054 g 116mM K 2 HPO 4 1.9032 g 84mM DoubledistilledH 2 OFillTo: 1300 mL Table2.3Theingredientsandamountaddedfor1mMacetatein2000mLtotalvolumeof medium. Ingredients Amount added Units Final concentration 10XDBStock 200 mL 100XDBMineralMix 20 mL 100XDLVitaminMix 20 mL 0.75MNaAcetate 2.67 mL 1 KH 2 PO 4 31.6 g 116mM K 2 HPO 4 29.28 g 84mM DoubledistilledH 2 OFillTo: 2000 mL 2.2.2Substrate Intheexperiment,3mMacetateconcentrationwasused.Thesubstratewasprepared (seeTable2.3)attheRegueralabatMSU.Inlatterexperiments,1mMacetateconcentration wasused(Table2.4). AftertheacetateandotheringredientswiththepHweremixedin2000mLtotal volume,pHwasrecordedas6.46.ThenNaOHwasaddedtosetthepHat6.73.Thenthe mediumwastransferredtotheside-armandvacuumedwithrapidstirringfor30min. Afterthevacuuming,itwassplittedinto200mLbottleswithpressuretubesandsparged withN 2 :CO 2 (80:20)withoutstoppersfor30minandthenwithstoppersforanother 30min.Thebottleswerecrimpedandautoclavedat121 Cfor30mininthedrymode. 39 2.3DataAcquisition Datawerecollectedfromapotentiostat(Omni-101Potentiostat,CypressSystems).Poten- tiostatsaredeviceswhichcontrolpotentialtogivetheinformationaboutthecharacteristics oftheelectrochemicaldevice(MFCinthiscase).Apotentiostatworkswiththreeelectrodes, theworkingelectrode(theanode),theAg/AgClreferenceelectrode,whichhasapo- tential,andtheauxiliary(counter)electrode(thecathode).Tomeasuretheopencircuit potentialoftheanode,therotarycontrolleronthepotentiostatshouldbeonstandby.To theanodepotentialatacertainvalue,thepotentiostatrotarycontrollerwasswitched tocell-onandthevalueoftheanodepotentialwassetat+0 : 240V.Thatway,thebacteria wouldbedriventoattachtotheanodeandformaEverythingwasmeasuredand displayedwithdSpace(ControlDesk5.1withCP1104board).Thecurrentwasmeasured throughacircuitbecausedSpace(Fig.2.10)acceptedonlyvoltageinputs. Figure2.8Thecircuitusedformeasuringthecurrent. TomeasurethepH(Fig.2.9),apHprobe(Vernier,USA)withaBNCconnectorwas usedandthedatawerecollectedateachtimeperiodtomakesurethatthepHwasstaying 40 ataround6.7allthetime.ThepHprobeshouldbecalibratedwithpHcapsuleseach timebeforeitisusedtogettheaccuratedata.SincedSpaceacceptsonlyvoltageinput, thevaluefromthepHelectrodetodSpaceneededtobeconvertedtothepHvaluethrough NernstequationasmentionedbeforeinSection2.1.5. Figure2.9ThepHprobewithBNCconnector. 2.3.1HPLC(HighPerformanceLiquidChromatography HPLCisamethodwidelyusedinanalyticalchemistry.Thismethodisahigherversion ofcolumnchromatography,whichisamethodusedforseparationofindividualchemical compoundsfromamixtureofcompounds.HPLCdoesthatwiththeaidofhighpressure, withamuchfasterwayofgettingtheresults.TheHPLCmethodwasusedtogetthe informationfortheacetateconcentrationinthesubstrate.Samplesweretakenfromthe MFCwithasterilesyringeandwastransferredtoacentrifugetube.Thesewerecentrifuged (theRegueralab)tospindownthebacteriainthesamples.Thesampleswerereservedina 41 Figure2.10dSpaceCP1104. freezeruntilthedaytheywereanalyzed.Acetateandmetabolicendproductsinthesample supernatantswereseparatedbyhigh-performanceliquidchromatography(Waters,Milford, MA)ona300by7.8mmAminexHPX-87Hcolumn(Bio-Rad,Hrcules,CA)at23 Cwith 4mMH 2 SO 4 astheeluent,atawrateof0.6mL/min[3]. 42 Chapter3 MicrobialFuelCellModeling Microbialfuelcells,todescribesimply,aredevicesthatgetenergyfromanaerobicdiges- tionofthemicroorganisms.Todevelopamodelforthisbehavior,oneshouldunderstand thebiologicalandelectrochemicaldynamicsunderlyingtheprocess.Themodelis importantforunderstandingtheelectrontransportationfromthebacteriatotheanodein mediator-lessMFCs;however,thesuspendedmicroorganismmodelisthefundamentaldevel- opmentforhowtheanaerobicdigestionworksandgivessimplerapproachforMFCmodeling andcontrol.Inthisstudy,themodelofsuspendedmicroorganismswillbeconsideredfor simplicity;however,thisapproachcanbeextendedtothemodel. 3.1MicrobialKinetics Theconnectionbetweentheactivebiomassandtheprimarysubstratesisthemostfundamen- talfactorneededforunderstanding.Becausethisconnectionmustbemadesystematically andquantitativelyforengineeringdesignandoperation,mass-balancemodelingisanessen- tialtool[16].Itisconsideredthatthelimitingfactorforbacterialgrowthisthesubstrate (electrondonor).Therelationbetweenthesetwodynamicsiswellknownandcapturedby theMonodequation.Here,foramoresystematicapproach,wewillconsiderthebacterial dynamicsincludingbothsynthesisanddecay.Intheliterature,usuallywhentheresearchers usebiomodel,insteadofthisapproach,onlysynthesispartofbacterialdynamicsis 43 consideredandthedecayrateisdesignedasthedetachmentratefromtheTherate ofsynthesiscanbewrittenas[1] syn = 1 X a dX a dt syn = max S K + S (3.1) X a istheexpressionfortheactivebiomass(fortheourmodel,however,thiswillbewritten as X asitrepresentsthetotalbiomassinthesystem,consideringthatthebacteriaare suspended),however,thesekineticscanbeapproximatedtodescribetheentirebacteria communityduetothenatureofsuspendedmicroorganisms.Inthisstudy,therewillbeno separationofactiveorinactivebiomassesandthenotationforallthesuspendedbiomassis X . Ontheotherhand,asmentionedabove,therewillbebacterialdecay.Studyingmore slowlygrowingbacteriahasshownthatactivebiomasshasanenergydemandformain- tenance,whichincludescellfunctionssuchasresynthesisandrepair,motility,transport, osmoticregulation,andheatloss[1].Environmentalengineersusuallyrepresentthatwof energyandelectronsrequiredtomeetmaintenanceneedsas endogenousdecay [16].Inother words,thebacteriaoxidizethemselvestomeetthoseneeds.Therateofendogenousdecay canberepresentedas: dec = 1 X a dX a dt decay = b (3.2) for b> 0,and b isendogenousdecaycot. Overall,thenetspgrowthrateofbiomass( )isthesumofgrowthanddecayrates: = 1 X dX dt = syn + dec = max S K + S b (3.3) 44 Thesubstrateutilizationisanotherkineticsworthtomention.Theratethatbacteria breakdownthesubstrateisalsorelatedtotheMonodequation.Whilethecellgrowthis derivedfromsubstrateutilization,theMonodequationtakestheform[1]: r s = q max S K + S X (3.4) where r s istherateofsubstrateconcentrationchange(substrateutilization). 3.2GrowthYield Thegrowthyieldisabiologicalvariablethatallowsustoassesstherateofelectron-donor electronsconvertedtobiomasselectronsduringsynthesisofnewbiomass.Substrateutiliza- tionandbiomassgrowthareconnectedby max = q max Y (3.5) where Y iscalledthegrowthyield. Thenetrateofcellgrowththenbecomes X s = Y q max S K + S X bX (3.6) inwhich, X s isthenetrateofbiomassgrowth.FormodelingMFCs,themeasuredgrowth yieldisbFrom(3.5), Y canbeinferredfromthetwovariables, q max and max . Butitisusefultoexplainthedirectmeasurementof Y .Wecandethegrowthyieldas therateofbacteriaconcentrationchangedividedbytherateofthesubstrateconcentration 45 change.Thenthegrowthyieldtakesaform, Y = r x r s (3.7) inwhich, r x istherateofbacteriaconcentrationchange(thenetgrowthrateofbiomass). Inbatchsystems,thegrowthyieldbecomes, Y = dX dt dS dt = dX dS (3.8) Here,theminussigncomesfromtheequation(3.4)sincetheconsumptionrateofthesub- strateisadecreasingchange. Foracontinuoussystem,tocalculatethegrowthyield,onecouldsimplyatime periodandtakethesamples(forbacteriaandsubstrateconcentrationschange)atinitial andterminaltimes,anddividingthemcouldgivetheapproximatevalueofthegrowthyield, Y = X S = X X 0 S S 0 (3.9) whereisthebacteriaconcentrationchangeinthetimeperiodandisthe substrateconcentrationchangeinthatsametimeperiod. X and S aretheterminal bacteriaconcentrationandsubstrateconcentration,respectively. X 0 and S 0 aretheinitial bacteriaconcentrationandsubstrateconcentration,respectively. 46 3.3PotentialofanMFC Thereisabetweenstandardandnon-standardelectromotiveforcesofanMFC. Ifthecellisunderstandardconditions,thecellpotentialcanbeobtainedbythe betweenthestandardelectromotiveforceofthecathodeandthestandardelectromotive forceoftheanodepartintheMFC.Standardconditionsarethosethattakeplaceat T =298.15[K] P =1[atm] M s =1.0[M] where T isthetemperature, P istheatmospherepressureand M s isthechemicalconcen- trationforliquidbasedontheIUPAC(InternationalUnionofPureandAppliedChemistry) convention. Non-standardconditionsoccurwhenanyofthesethreeconditionsisviolated,butgen- erallytheyinvolveachangeinconcentration.TheOCPofMFCsarebasedontheNernst- Monodequation.Nernst-Monodequationisaquantitativeexpressionthatdescribesthe relationshipbetweentherateofEDutilizationandtwovariables:EDconcentrationand electricalpotential[35].Thestandardpotentialofchemicalreactionscanbefoundinthe literature[1],[18]. 3.3.1ThermodynamicAnalysis ThereactionsoccurringinanMFCcanbeanalyzedintermsofthehalf-cellreactions,or theseparatereactionsoccurringattheanodeandthecathode[18].Forexample,theacetate 47 oxidizationintheanodecompartmentcanberepresentedas CH 3 COO +4 H 2 O = ) 2 HCO 3 +9 H + +8 e (3.10) whereasonthecathodeside, O 2 +4 H + +4 e = ) 2 H 2 O (3.11) 3.3.1.1Activity Inchemicalthermodynamics,activityisameasureofthectiveconcentrationofspecies inamixture,inthesensethatthespecies'chemicalpotentialdependsontheactivityofa realsolutioninthesamewaythatitwoulddependontheconcentrationforanidealsolution [16]. Activityofasubstanceisadimensionlessvalue.Theactivityofpuresubstancesin condensedphases(solidsorliquids)isnormallytakenasunity(=1)[36].Forareaction[36], aA + bB $ cC + dD ,wheretheleft-handsiderepresentsthereactantsandtheright-hand siderepresentstheproducts,thereactionquotienthastheform: Q = [ C ] c [ D ] d [ A ] a [ B ] b (3.12) where[ C ]isunderstoodtobethemolarconcentrationofproduct C ifitisaqueous,orthe partialpressureinatmosphereifitisagas. Basedonthechemicalreactioninvolvingthesubstrateontheanodeside,wecancalculate 48 thepotentialontheanodeside E Anode : E Anode = E 0 a RT nF ln( Q )(3.13) inwhich, E 0 a isthestandardpotentialontheanodeside.Standardpotentialisthepotential underthestandardconditionsofthosethatwereexplainedearlierinthissection. R , T , n , F aretheidealgasconstant,thetemperatureofMFC,thenumberofelectronstransferredand theFaradayconstant,respectively.Thereactionquotientforthechemicalreaction(3.10) whichhappensontheanode,willbe, Q = [ CH 3 COO ] ([ HCO 3 ]) 2 ([ H + ]) 9 (3.14) where CH 3 COO , HCO 3 and H + representtheacetateconcentration,thebicarbonate ionconcentration,andtheprotonconcentration,respectively.Inthesamemanner,wecan calculatethepotentialonthecathodesidebasedonthechemicalreactionthere, E Cathode = E 0 c RT nF ln( Q )(3.15) inwhich, E 0 c isthestandardpotentialonthecathodeside. Thus,theoverallpotentialofanMFCcanbecalculatedas, E Cell = E Cathode E Anode (3.16) 49 3.4VoltageOutput Intheprevioussection,thepotentialsdescribedintheEqs.(3.13)-(3.16)aretheoretical calculationsdependingonthechemicalreactionshappeningintheMFC.Themeasured voltageofanMFCwillbelowerthanthisvalueduetothepotentiallosses,asexplained below. 3.4.1ActivationLosses Theactivationlossoccursatthebeginningoftheprocessofthesystem.Becauseachemical reactionneedstobeactivatedtostart,activationlossoccursduringthetransferofthe electronsfromthebacteriatotheanode.Theactivationlossateachelectrodeofafuelcell isgovernedbytheButler-Volmerequation[49], I MFC = i 0 A sur [exp( 1 nFV act RT ) exp( 2 nFV act RT )](3.17) where I MFC istheMFCcurrent, i 0 istheexchangecurrentdensityinreferenceconditions, A sur istheanodesurfacearea,and V act istheactivationloss(thelossisintermsofpotential sinceitistakenfromthetotalvoltageontheelectrode)ontheanode.Thereduction( 1 ) andoxidation( 2 )transfercotsaredeterminedbytheelectrontransferprocessesat theelectrode-electrolyteinterface[39].Thesecotsaredirectlyrelatedtotheelectrode reactionmechanismandaretoidentify[49].Theexchangecurrentdensityinref- erenceconditionsisastrongfunctionofelectrodematerials,design,reactantandproduct concentrations,andtemperature[39].Eq.(3.17)canbereducedtotheTafelequationfor 50 largevaluesofactivationlosses, V act ˇ RT nF ln( I MFC i 0ct A sur )(3.18) Forsmallvaluesof V act ,Eq.(3.17)canbereducedtoalinearrelationshipbetween thecurrentandtheactivationloss.Thisreductionisoftencalled\linearcurrent-potential equation"asshownin[60],[39], V act ˇ RT nF ( I MFC i 0 A sur )(3.19) Itmustbenotedthat,[49]hasclearlydemonstratedthatButler-Volmerapproximations leadingtotheTafelandlinearcurrent-potentialequationsshouldbecautiouslyusedin modellingandmodelanalysis,becausetheycouldntlydeviatefromtheButler- Volmerequationoutsidetheirrangeofapplicability[39]. 3.4.2OhmicLosses Resistancetothewofelectronsandionsduringthefuelcelloperationgeneratesohmic losses.Theselossesincreaseasthecurrentwincreasesandthislinearrelationshipobeys Ohm'slaw;therefore,ohmiclossescanbedescribedby[6]: V ohm = R int I MFC (3.20) where V ohm and R int areohmiclossontheanodesideandtheinternalresistanceofMFC, respectively. Ohmiclossesarisefromresistanceofion(proton)conductionduetothesolutionand 51 (ifpresent)themembrane,andresistanceofthewofelectronsfromtheelectrodetothe contactpoint(i.e.,wheretheelectrodesareconnectedtoawire),andanyrelevantinternal connections.Ohmiclossescanbelimitedbyreducingelectrodespacing,choosingmembranes orelectrodecoatingswithlowresistances(ifpresent),ensuringgoodcontactsbetweenthe circuitandelectrodes,andincreasingsolutionconductivityandcapacity[18]. 3.4.3ConcentrationLosses Theconcentrationsofthereactantsandproductsinthefuelcellatthecompartmentbulk phaseareoftentfromtheirconcentrationvaluesattheelectrodesurface.Dueto consumptionandformationreactions,reactantsaresparseattheelectrodesurface,while productsareabundant.Thisconcentrationgradientleadstoamasstransportphenomenon thatisdeterminedbySincethecurrentproducedbythefuelcellislinkedtothe electrodereactions,thesionofreactantsandproductsthefuelcellperformance. Thisiscalledconcentrationlosses[39]. Theconcentrationlossescontributetlytothedecreaseincellpotential,partic- ularlyathighcurrentdensitiesandlowbulkreactantconcentrations[49],[39].Theselosses canbedeterminedbythepotential E )betweenthevoltageatopencircuit(bulk concentration, E i =0 )andthecellvoltageathighcurrentrates(E i high )[6].So,theNernst equationcanbeappliedbetweenthereactants'concentrationsinthebulkliquid( C Bulk )and ontheelectrodesurface( C Surface )as: E = V conc = RT nF ln( C Bulk C Surface )(3.21) Inaddition,onecan I R L asthelimitingreferencecurrent,e.g.,themaximum 52 possiblecurrentdensity,atwhichthemaximumrateofreactantscanbesuppliedtothe electrode.Bythisn, C Surface iszeroat I R L .NowbyapplyingFick'slawatthe limitingreferencecurrentandbyusingeq.3.21onecan[6],[39], C Bulk C Surface = I R L I MFC I R L (3.22) Therefore,theconcentrationlossescanbewrittenasafunctionoffuelcellcurrentand itslimitingreferencecurrent[39], V conc = RT nF ln(1 I MFC I R L )(3.23) 3.4.4TheVoltageOutput Basedonthepreviousdiscussions,themeasuredpotentialontheanodeelectrode, V Anode = E Anode [ V act + V ohm + V conc ](3.24) Forthecathodepotential,thesameprocedurecanbeapplied;however,inthisstudy,the cathodepotentialisassumedtobeconstant.Theassumptionofconstantcathodepotential comesfromthemainattentionontheanodepotentialoutput[48]. 3.5AControl-orientedModelforMFCs Inthissection,themathematicalmodeldevelopedforthepurposeofcontrollerdesignis summarizedinaformthatclearlyindicatesstatevariables,controlinput,andoutputs.A controlvolumeisdescribedasamathematicalabstractiontohelpbuildthemathematical 53 modelsofphysicalprocesses(chemicalreactionsinthisstudy).Inthecontinuous(orfed- batch)mode,MFCscanbeperceivedaschemostats(acontrolvolumewhichincludesthe organicmatterandthemicroorganism).Inthischemostattheinputandtheoutputare controlledforthedesiredpurposes.Theproposedmodelconsidersonlyonekindofmicroor- ganismandusesacetateasthesubstrate,buttheapproachisamenabletogeneralizationto amixtureofmicroorganismsandothertypesofsubstrates.Thefollowingassumptionsare madeinthemodeldevelopment: 1.Themixingofsubstrateisideal,andthesubstrategradientintheisneglected. 2.thesubstrateconcentrationchangeisthemainontheanodeOCPoutput. 3.Thetemperatureremainsconstantattheroomtemperature,andthepHiskeptcon- stantviathepHinthemedium. 4.Themainoverpotentialthecathodepotentialistheactivationloss.For andbecauseofthesmallchangesinthecathodeOCP,thecathodeOCP isassumedconstant[48]. 5.Thereisnomanualadditionofactivebiomasstothesystem. 3.5.1MassBalances Describingmassbalancesandtherateofsubstrateconcentrationchangerequiresspecifying acontrolvolume.InSection3.1,theconcentrationsofbacteriaandthesubstrateinthe batchmode(closedvolume)arediscussed.Inthissection,theinputtothecontrolvolume isincludedinthestateequations.TheliquidvolumeoftheMFCisdenotedas V c .The systemreceivesafeedwwithrate F c ,havinginitialsubstrateconcentrationof S 0 ,which 54 describedastheMFCinitialsubstrateconcentrationbefore.Thissubstratefeediscoming fromthesubstratestoragecontainer(Fig.2.1).Thestoragecontainerhasthesubstrate concentrationof S 0 ,soitfeedstheMFCcontainerwiththatvalue.Therateofchangein substrateconcentrationintheMFCis, dS dt = qX + D ( S 0 S )(3.25) Thedilutionrate, D ,isthecontrolinputtothesystem.Bychangingthevalueof D ,onecan examinethewrateontheoutputoftheMFCsincethedilutionrateisafunction ofthewrate, D = F m V 1 m (3.26) InEq.(3.25), q isrelatedto r s in(3.4), q = q max S K + S (3.27) Inthecontinuous(orfed-batch)system,thebiomassmassbalanceequationis, dX dt = DX (3.28) Again,theparameter representsthenetspgrowthrateofthebacteria. 55 3.5.2SummaryoftheControl-orientedModel 3.5.2.1StateEquations TofacilitatethediscussionontheanalysisoftheMFCdynamics,weintroducenewnotation forthedynamicsthatismorecommonlyusedinthecontrolliterature.Let x 1 represent thesubstrateconcentration(theoriginal S )and x 2 representthebiomassconcentration(the original X ).Theinput, u ,representsthedilutionrate, D .Thesystemdynamicscanbe representedas _ x 1 = q max x 1 K + x 1 x 2 + u ( S 0 x 1 )(3.29) _ x 2 =[ max x 1 K + x 1 b u ] x 2 (3.30) _ x 3 =2( q max x 1 K + x 1 x 2 u ( S 0 x 1 ))(3.31) _ x 4 =9( q max x 1 K + x 1 x 2 u ( S 0 x 1 ))(3.32) wherenewstatevariables x 3 and x 4 arethenewrepresentationsof HCO 3 and H + con- centrations,respectively.Theoftheseconcentrationsinthevoltageoutputis explainedinSection3.3.1.1.Thereasonforthesestatevariableschosenthatway(Eqs. (3.31),(3.32))isduetothechemicalreactionthathappensontheanodeside;see(3.10). Thechangeoftheactivityfor x 3 and x 4 isassumednottotheacetateconcentration equation,becausethesystemisclosedandthereisnotransferofionsinoroutofthesystem. 56 3.5.2.2Output DependingontheoftheMFCmeasurementsetup,therearemultipleways forthesystemoutput.Beforewritingthecontrol-orientedOCPoutput,allthe parametersintheequationshouldbewrittenintermsofstatevariables,constantparameters andtheinputs.Onecommonandconvenientchoicefortheoutputistheopen-circuitvoltage: V Anode = E 0 a RT nF ln( x 1 = ( x 2 3 x 9 4 )) [ RT nF ( I MFC i 0 A sur ) + R int I MFC + RT nF ln(1 I MFC I R L )](3.33) ThisequationcomesfromSection3.4withactivation,ohmicandconcentrationlosses subtractedfromthetheoreticallycalculatedOCPofanode. Tobeabletoestimatetheinternalresistance,thefollowingequationcanbeused[39], R int = R MIN +( R MAX R MIN ) e K R X (3.34) Again, X isthebiomassconcentrationandwillbedenotedas x 2 asmentionedbefore.. Anothervariablethatshouldbewrittenintermsofthestatevariablesis I MFC .Inthe study[37],theauthorsthecurrentdensity: j = j max S K + S (3.35) where S and K arethesameasbefore,thesubstrateconcentrationandtheconcentration givingone-halfthemaximumrate,respectively, j isthecurrentdensityontheanodepart 57 andj max isthemaximumcurrentdensityobtainedontheanodepart.This however,wasbasedontheregionofbulksubstrateconcentrationinamodel.So, thisisapproximatedinthisstudyforthebulksubstrateconcentration.Sincethe currentdensityisthecurrentdividedbythesurfacearea(anodesurfaceareainthiscase), thenthecurrentbecomes, I MFC = A sur j max S K + S (3.36) or, I MFC = A sur j max x 1 K + x 1 (3.37) ThenEq.(3.33)becomes, V Anode = E 0 a RT nF ln( x 1 = ( x 2 3 x 9 4 )) [ RT nF ( j max x 1 K + x 1 i 0 ) +( R MIN +( R MAX R MIN ) e K R x 2 ) A sur j max x 1 K + x 1 + RT nF ln(1 A sur j max x 1 K + x 1 I R L )](3.38) 3.6ExperimentalModelIden 3.6.1OCPMeasurement Modelidenwasconductedusingthedatafromtheexperimentsforboththe casesof3mMand1mMacetateoperatedinthebatchmodeandfromtheHPLCresults. Matlabsimulationresultswereusedtoouttheintervalsfortheestimated parameters.Forthedatacollection,theMFCwasinoculatedwith3mMacetateand 10mLofbacteriaculturewithaconcentrationof5x10 10 cells/mLinitially.Themaximum 58 opencircuitpotential(OCP)oftheanodewasobservedtobe 471mV(Fig.3.1)andits magnitudewasdecreasingafterwards(Fig.3.2). Figure3.1AnodeopencircuitpotentialversusAg/AgClreferenceelectrodeafterthebacteria inoculation. Forthesecondexperiment,theMFCwasinoculatedwith1mMacetateandthesame volumeanddensityofbacteriainitially.ThemaximumOCPoftheanodewasobservedto be 362mVandsimilarlyitwasdecreasingafterwards.Theidenparametersfrom theOCPandHPLCresultswere, max and q max .Therearesomeparametersthatare 59 toidentify,forwhichwehaveusedthedatafromtheliteraturetodeterminetheir values. Figure3.2MaximumpointofanodeOCPversusAg/AgClreferenceelectrode.TheOCPof anodewasdecreasingafterthatpoint. Fortheestimationofthoseidenparameters,wascarriedout.Basically, thebestvaluesfortheparameterswerechosenbasedonthevaluesgatheredfromtheOCP valuesforthesubstrateconcentration( x 1 ).Thisvaluethenwasalsovalidatedwiththe HPLCanalysisresults.Becauserunningtoomanyexperimentswasnotavailableinthis study,thedatafromtheliteraturewasusedtoidentifytheotherparameters(onesthatare notidenwiththecurrentexperimentaldata).Theidenofseveralparameters presentintheOCPequationrequiresadditionalexperimentssuchascyclicvoltammetryand polarizationtestswithtexternalresistors;however,inthisstudy,theseexperiments werenotconductedyet.TheparametersestimatedforoursystemarelistedinTable3.1. 60 3.6.2HPLCMeasurement Theacetateconcentrationinitiallywas1mM(59mg/L).Thisinitialconditionwasused inMatlab/Simulinksimulationofthemodel.Simulationsuggeststhatalmost63%ofthe acetatewasconsumed(Fig.3.3)bythebacteriain6daysandthatresultagreedwiththe HPLCresultwhichwastakenafter6daystheinoculationwasdone.TheHPLCresults showedthattheacetateconcentrationwas0.38mMon6thday. Figure3.3Thesimulationresultsfortheacetateconcentrationchangewithzeroinput(batch mode). Theparameterssuchas R ,and F areuniversalconstantswithknownvalues.The valuefor T ,wastakenas298.15KwhichisthetemperatureoftheroomintheMFCand isassumedtostayunchanged.Theanodesurfaceareawascalculatedapproximatelyby measuringthedimensionsoftheanodebrush(0.67m 2 ).Theelectronsperchemicalreaction is8e whichisknownfrom(3.10).Thevaluefor E 0 a (0.187mV),whichistheanode 61 Table3.1ParametersidenfortheMFCmodel. Parameter Description Value Unit Explanation q max maximumsp rateofsubstrate utilization 3 day 1 estimated K concentration givingone-half themaximumrate 27 mgL 1 assumed max maximumsp growthrate 0.5 day 1 estimated S 0 initialacetate concentration 60 mg/L known X 0 initialbacteria concentration 1.5 mg/L known F Faradayconstant 96485 sA/mol constant R idealgasconstant 8.31446 JK 1 mol 1 constant T MFCtemperature 298.15 K constant n numberofelectrons transferred 8 dimensionless known b endogenousdecay cot 0.07 day 1 estimated E 0 a standardanode potential 0.187 V constant A sur Anodesurfacearea 0.67 m 2 calculated OCPunderthestandardconditions,wastakenfromtheliterature[4],[18].Thisvaluehas alreadybeencalculatedbefore,accordingtothechemicalreaction(3.10)thathappensin theanodepartwiththeacetateasthesubstrate.Thevaluesfor max ,b(theendogenous decaycot),and q max wereestimatedbyusingtheOCPandHPLCdatafromthe experimentsviadatamethod(Table3.1).ThevalueforKistheleastpredictable value.Thisnon-idenparameterwastakenfromtheliterature[1],[35],[58]basedon thesimilaroperatingconditions. 62 3.6.3pHmeasurement ForthepHmeasurement,thepHelectrodewiththeBNCconnectorwasused.Thesamples forthepHvaluesweretakenfromtheMFCbottlewithasterilizedsyringetwiceeveryday withapproximately12hoursintervals.ThecalibrationforthepHelectrodewasmade manuallyasdescribedbeforeinSection2.1.5.1.ThecalibrationforthepHelectrodewas neededeveryday.AsitcanbeseeninFig.3.4,thepHdidnotchangetlydueto thepHinthesolution(potassiumphosphate). Figure3.4ThepHvaluestakentwiceaday. 3.7ModelValidation Modelvalidationhasbeendoneusing1mMacetateconcentrationandthe5x10 10 cells/mL, andwehad10mLbacteriaculturewhichmeansthattherewas5x10 11 cellsinitiallyinthe 63 MFC.Forthesimulationpart,the x 3 and x 4 initialvaluesweretakenaszerosincetheyare assumedtobeabsentuntilthebacteriastartsconsumingorganicmatter.Thevaluesfor the x 3 and x 4 arelinearlydependenton x 1 inthemodelproposedinthisstudy.Withthis simulation,theresultfor1mMacetateconcentrationisshowninFig.3.5. Figure3.5SimulationresultsfortheanodeopencircuitpotentialversusAg/AgClreference electrodeafterthebacteriainoculation. TheMFChad220mLmediumbubblingwithN 2 for5hoursbeforethebacteriainocula- tion.TheOCPoftheanodewasobservedtobe+112mVatthattime.Then10mLofthe bacteriainoculationwasaddedandtheOCPofanodestartedincreasingdrastically.The OCPthatwasobservedontheanodesidewas 380mVin2.5hoursafterthebacteriawere added(Fig.3.1).Thentheobservationshowedthatin60hoursthemagnitudeofOCPof theanodewentdownto 320mV(Fig.3.6).Notethat,becauseofthedSpaceconverting thevoltagevaluesignfromthepotantiostat,thesignsofthevoltagevaluesseemtohave 64 ed. Figure3.6AnodeopencircuitpotentialversusAg/AgClreferenceelectrodeafterthebacteria inoculation. Basedonthemodel,wehaveconductedadditionalsimulationanalysis.Fortheinitial bacteriaconcentration,ithasbeenestimatedtobe5x10 13 cells/mL.Withonebacterium massweightof3x10 17 g[64],theinitialbacteriamassconcentrationfollowinginoculation wasestimatedtobe1.5mg/L(Fig.3.7).Thebacteriaconcentrationreachedthemaximum levelin7daysanditstarteddecreasingduetothedepletingacetateconcentration.However, simulationresultsshowedthatthebacteriaconcentrationdidnotreachtozeroevenin20 daysafterthesubstrateconcentrationwasalmostzero.Itisbecausethebacterialendogenous decaycotintheEq.(3.7)wassmall. 65 Figure3.7Thesimulationresultsforthebacteriaconcentrationchangewithzeroinput (batchmode). 66 Chapter4 MFCModelAnalysis Inthischapter,theMFCmodelwillbeanalyzedinthefed-batch(whereeither u =0ora constant)andcontinuousmode( u canbechanged).Forthecontinuousmode,werestrictto thecaseof u beingconstant.First,theequilibriaofthesystemwillbeinvestigatedandthe propertiesoftheseequilibriumpointswillbestudied.Wewillfurtherexplorethebifurcation ofthesystembehaviorasthe(quasi)constantcontrolinput u increases.Stabilityofthese equilibriawillalsobeanalyzed. 4.1EquilibriaoftheSystem ItisimportanttounderstandthepropertiesofthesystemdynamicsinanMFC,whichwill beinstrumentalintheMFCcontrolandoptimization.Oneimportantpropertyistheset ofequilibriaandtheirstability.Recallthestateequations(3.29-3.32).Thetwostate equationswillbeconsideredinthischaptertoanalyzethebehaviorofthesubstrateandthe biomassconcentration. Fromthesecondequationofthestateequations,thereisatrivialequilibriumpoint, x 2 =0 ) x 1 = S 0 (4.1) whichmeansthatifthereisnobacteria,thesubstrateconcentrationwillremainconstant 67 regardlessoftheinputvalue.Theotherequilibriumpointiswhen x 1 , x 1 = K ( b + u ) max b u (4.2) thenthecorresponding x 2 is x 2 = max u ( Kb + S 0 b + Ku S 0 max + S 0 u ) q max ( b + u )( b max + u ) (4.3) Thelatteristhenontrivialequilibriumpoint,whichdependsontheinputvalue.The importanceofthedilutionrate(theinput)couldbeseeninthisequilibriuminacontinuous- modeMFC.Fromthisequilibriumexpression,wenotethatif u = max b ,no equilibriumcanbeachieved. Aspecialcaseofinterestiswhen u =0(thefed-batchmode).Inthatcase,oneofthe eigenvaluesbecomeszero.Whenoneorbotheigenvaluesarezero,thephaseportrait(the familyofalltrajectories)isinsomesensedegenerate[2].Whenthatsituationhappens, thereisanullspace,anyvectorinwhichisanequilibriumpointforthesystem;thatis, thesystemhasanequilibriumsubspace,ratherthananequilibriumpoint.InFig.4.1,this subspacecanbeseenwiththeredmarksonthephaseportrait.Dependingontheinitial conditionsforthebacteriaandthesubstrateconcentrations,anequilibriumwillbereached intheequilibriumsubspace.Inthiscase,theequilibriumspaceis x 2 =0and x 1 cantake anypositivenumberbutK(concentrationgivingone-halfthemaximumrate). 68 Figure4.1Thephaseportraitfortheequilibriumsubspacewithzeroinput(fed-batchmode). 69 4.2StabilityoftheEquilibria Considerthenonlineartime-invariantsystem(Eqs.(3.29-3.32)).Weanalyzethelocal stabilitythroughlinearizationattheequilibriumpointofinterest.Inparticular,theJacobian matrixis, J = 2 6 6 6 6 6 6 4 Kq max x 2 ( K + x 1 ) 2 u q max x 1 K + x 1 K max x 2 ( K + x 1 ) 2 max x 1 K + x 1 b u 3 7 7 7 7 7 7 5 (4.4) Atthetrivialequilibriumpoint, x 2 =0and x 1 =60,withthevaluesfromtheTable 3.1,the J matrixis, J = @f @x = 2 6 6 6 6 6 4 u 2 : 069 00 : 275 u 3 7 7 7 7 7 5 (4.5) Since u 2 R + ,unless u 0 : 275,theequilibriumpointislocallyasymptoticallystable becauseboththeeigenvaluesarerealandnegative.However,somethinginterestinghappens whenthedilutionratereachesonespvalue,when u =0 : 275,oneoftheeigenvalues 1 = 11 = 40andtheothereigenvalue 2 =0,whichgivesanothernullspace.Ontheother hand,when u< 0 : 275,oneofthetwoeigenvaluesisnegativewhiletheotherispositive.In thatcasethisequilibriumpointiscalledasaddlepointanditisnotstable. ThesecondequilibriumpointofEqs.(4.2)-(4.3)ismoreimportantsinceitismore 70 practicallyrelevant.The J matrixforaconstant u becomes, J = 2 6 6 6 6 6 4 u (290000 u 2 159400 u +37421) 450(100 u +7) 6 u 21 50 u (290000 u 2 204400 u +34271) 2700(100 u +7) 0 3 7 7 7 7 7 5 (4.6) Forthismatrix,theeigenvaluesasafunctionof u arecalculated, 1 = (37421 u +( u (84100000000 u 5 144652000000 u 4 +76596540000 u 3 13203494800 u 2 +716982841 u 30227022)) 1 = 2 159400 u 2 +290000 u 3 ) = (900(100 u +7))(4.7) 2 = (37421 u ( u (84100000000 u 5 144652000000 u 4 +76596540000 u 3 13203494800 u 2 +716982841 u 30227022)) 1 = 2 159400 u 2 +290000 u 3 ) = (900(100 u +7))(4.8) UsingMatlab,theeigenvaluesforincreasing u fromzeroto0.3by0.01incrementcalculated. Theplottedeigenvaluesfortheset u valuesareasshowninFig.4.2. Aninterestingphenomenum,calledbifurcation,isimportantinthepracticalsense.It iswhetherthesystemmaintainsitsqualitativebehaviorundersimallysmallpertur- bations[2].Inthiscontrolmodeltheinputchangestoaspvalueandthenon-trivial equilibriumpointchangesitsstabilitypropertyatacertaininputvalue.Withtheparame- tersinTable3.1,thisinputvalueis u =0 : 275.Thisisthesameinputvaluewiththeprevious case.Thatinterestingresultshowsusthatboththesetwoequilibriumpoints(Eqs.(4.1) 71 Figure4.2Theeigenvaluesfortuvaluestoshowthebifurcationpoint.Theorange lineshowsthetrajectoryoftheeigenvalueandthepurplelineshowsthetrajectoryof thesecondeigenvalue. 72 and(4.2,4.3)),arechangingtheirpropertyatthatsp u value.Ourinterestismainly ontheequilibriumpoint(Eq.(4.1))sincethesecondequilibriumpointgoesbeyondthe physicaldomain.Theequilibriumpointbecomeslocallyasymptoticallystable.This resultsuggeststhatafterthesp u value,thewashoutwillhappen.InFig.4.2,until aroundthevalue u =0 : 2,eigenvaluesarecomplexandinthegraph,onlytherealpartof theeigenvaluesisconsideredsinceitisdecidingthestabilityproperty.Afterthat u value, thereisanother u value, u =0 : 275,atwhich,itiseasytoseethattheeigenvalueis switchingitssign(whiletheotherremainsatnegative),whichcausesabifurcationatthat point(fromstabletosaddle). Thelinearizedsystematthesecondequilibriumpoint(Eq.(4.6))strictlydependson u . For0