Oscillatory Chemical Reaction in a CSTR with Feedback Contro(3)

Figure4cillustratesthetransientpartafterkmax,1andkmax,2aresettoone-halfoftheirpreviousvalue(comparewithFigure4b).Becausetheflowratek01islargerthanthenewlysetparameterkmax,1,k01decreasesevenforhigh[I-]inthereactor.Afterseveralbursts,theflowratek01dropsbelowkmax,1.Theflowratek01cannotexceedtheparameterkmax,1butmustdecreaseiftheinitialvalueofk01isabovekmax,1.Theratiokmax,1/kmax,2playsamoresignificantroleinthedynamicsthantheabsolutevaluesofkmax,1andkmax,2.Whenwechangekmax,1andkmax,2whilemaintainingtheirratio,theburstingpersistswithonlyslightchanges(seeFigure4b,c).NumericalSection

Model.ThemodeloftheCDIreactionproposedbyLengyeletal.15isbasedontwooverallstoichiometricreactions:

ClO-fClO-1

2+I2+2

2

R1)k1XY

ClO-+2+4I-+4HfCl-+2I2+2H2O

R2)k2aZYH+

k2bZPYu+Y

2

(3)

HereX)[ClO2],Y)[I-],Z)[ClO2-],P)[I2],H)[H+],andRirepresenttheratelaws.Therateconstantsandparametersusedinthesimulationsarek1)6000M-1s-1,k2a)460M-2s-1,k2b)2.55×10-3s-1,u)1×10-14M2.ForaCSTRwithaniodide-dependentflowrateofallinputreagents(setupA,Figure1a)thesystemofdifferentialequationsis

J.Phys.Chem.A,Vol.101,No.28,19975151

dX

dt

)-R1+k0(X0-X)dY

dt

)-R1-4R2+k0(Y0-Y)dZ

dt

)R1-R2-k0Z(4)

τdk0

dt)kmax-k1+(Y/I0T)n

-Y

P)

Y02

Herek0isthedynamicflowrate,whichvariesaccordingtotheconcentrationofiodideinthereactor,andX0andY0aretheinputconcentrationsofXandY.ThefeedbackmechanismcontrolsthespeedoftheinputreagentsflowintotheCSTRwhilekeepingtheinputconcentrationsconstant.BecausetheinputconcentrationofH+ishighincomparisonwiththatoftheremainingspecies,wecanneglectprotonconsumptionineq3andassumethat[H+]insidethereactorisfixedatitsinitialconcentration.

InsetupB(Figure1b)theinputconcentrationsofallspeciesvarywiththeratioofthefixedflowratek01tothedynamicalflowratek02,andwemustconsider[H+]tobeadynamicalvariableaswell.TheequationsforsetupBare

dX

dt

)-R1+k01Xs-k0XdY

dt

)-R1-4R2+k02Ys-k0YdZ

dt

)R1-R2-k0ZdH

dt

)-4R2+k01Hs-k0H(5)

τdk02dt)kmax,2-k1+(Y/I02T)n

P)

k02Ys-k0Y

2k0

k0)k01+k02

Heresubscriptsindicatesconcentrationofthestocksolutions,k01isthefixedflowrateofClO2andH2SO4,andk02isthedynamicalflowrateofiodide.

ThemathematicalmodelofsetupCisdescribedbytheequationsforsetupB(eq5)supplementedwiththeequationforthepositivefeedbackofflowratek01:

τdk01

kmax,1dt)-k1+(I/Y)n

01(6)

T

Toanalyzeeqs4,5,and6,weusedtheCONTnumerical

bifurcationandcontinuationpackage.18Periodicsolutionswereobtainedbynumericallyintegratingthesystemofordinarydifferentialequations.Theintegrationusedasemiimplicit

5152J.Phys.Chem.A,Vol.101,No.28,1997Figure4.DynamicalbehaviorinsetupCexperiments:iodideandflowratetimeseries.(a)Maximumflowrateskmax,1)kmax,2)0.00833s-1,targetiodideconcentrationIT)1×10-6M.(b)kmax,1)0.00833s-1,kmax,2)0.01666s-1,IT)1×10-5M(firsthalf),1×10-6(secondhalf).(c)kmax,1)0.00416s-1,kmax,2)0.00833s-1,IT)1×10-6

M.

fourth-orderRunge-Kuttamethodwithautomaticcontrolofthestepsize.

ResultsofSimulations

ThemodeloftheCDIreactioninaCSTRwithoutfeedbackcontrolshowsgoodagreementwithexperiments.8,9,16Previoussimulations14withthemodelofsetupArevealedthepresenceofburstingandchaos.TheseresultsinspiredtheexperimentalinvestigationoftheCDIreactionwithfeedbackcontrol.WeperformedadditionalnumericalsimulationstoinvestigatethedynamicsofsetupsA,B,andC.Modelsofthesesetupsshowburstinginsimulations(Figure5).Shownherearetheiodideconcentration(pI)-log10[I-])andthedynamicallychangingflowrates.Figure5adisplaysburstingwithfivespikesperburstinsetupA.Duringspiking,theflowrateincreasessharply,andthemaximumflowratereachesnearly3timesitsminimumvalue.Theflowrateincreaseisfollowedbyaquiescentperiodduringwhichtheflowratedecreaseisnotassharpasthepreviousincrease.BurstinginsetupBshowsmuchsmallervariationsoftheflowrate(Figure5b).InsetupC(Figure5c)theoppositesignsoftheflowratevariationsresultinanalmostconstanttotalflowrate.ThespikinginsetupsBandChashigherfrequencythaninsetupA,asobservedinourexperi-ments.

Wehavestudiedindetailthedynamicsofallsetupsasafunctionofthetargetconcentrationandthecontrolparameterkmax.Figure6displaysthebifurcationdiagramofdynamicregimesforsetupAasafunctionofthetargetconcentrationITwhenkmaxisfixedat0.05s-1.ThediagramisobtainedfromaPoincare′map,withtheflowratek0evaluatedat[I-])1×10-7M,fordecreasing[I-].Figure6ashowsafulldiagramwithprevailingburstingbehavior.Thenumberofoscillationsperburstdecreaseswithincreasingtargetconcentration.Simplelow-frequencyperiod-oneoscillationsemergejustbeforethe

Dolniketal.

Figure5.Burstingoscillations,simulations.(a)SetupA:targetconcentrationIT)1×10-5M,maximumflowratekmax)0.05s-1.(b)SetupB:IT)1×10-5M,kmax)0.0166s-1,fixedflowratek01)0.0025s-1.(c)SetupC:IT)1×10-5M,kmax,1)0.0083s-1,kmax,2)0.0166s-1.

highvalueofthetargetconcentrationstabilizesthesystemattheHIsteadystate.WefindsimilarscenariosforothervaluesofkmaxandalsoformodelsofsetupsBandC.Inmostsimulationsthechaoticoscillationsarerestrictedtoarelativelynarrowparametricregion,whileregularburstingandsimpleperiodicoscillationsarewidelypresent.Figure6bshowsablowupofthediagramwiththeperioddoublingsequencesleadingtochaos.Thechaoticregioncontainsnarrowperiodicwindows,includingperiod-threeandperiod-fivewindowswiththeirownperioddoublingsequences.Burstingchaoticoscil-lationsemergeatIT≈9×10-7MwithincreasingIT.AsITincreasesfurther,theburstingoscillationsbecomeperiodic.WesummarizetheresultsofsimulationsforsetupsA,B,andCinFigures7and8.Inthesefigures,thesolidlinesconfinetheentireoscillatoryregionsandthedashedlinesrepresentsubcriticalHopfbifurcationlinesobtainedfrombifurcationanalysis.Outsidethesolidlinesthecontrolmechanismalwaysstabilizesthesteadystate.Thesteadystatecanalsobestabilizedforspecificinitialconditionsoutsidethedashedlines.Shadedareasrepresentbursting;greylevelsillustratethenumberofspikesperburst.Theblackregionisthedomainofchaoticoscillations.Figure7showsthatburstingisfoundonlyforintermediatevaluesofkmaxinallsetups.Thedynamicsinthesesetupsisqualitativelythesame;fromaquantitativepointofviewthereisbetteragreementinthelocationoftheburstingregionbetweensetupsBandC(Figure7b,c).Theshapeoftheshadedareassuggeststhatthedynamicsisalmostinde-pendentoftargetconcentrationwhenIT∈(1×10-7,1×10-5)s-1,whilethestrongdependenceonkmaxisapparent.Figure8demonstratesthecloseresemblancebetweensetupsBandC.Thefixedflowratek01insetupBplaysnearlythesameroleasthemaximumflowratekmax,1insetupC.Thevariations

of

OscillatoryChemicalReactionFigure6.One-parameterbifurcationdiagramssimulationsforsetup

A.Pointsinthediagramcorrespondtovaluesofflowratek0onPoincare

′surface:Y)1×10-7M,kmax)0.05s-1

.Smallnumbersofpointscorrespondtoperiodicoscillations,whilemanypointsindicatechaos.(a)Fulldiagramwithprevailingburstingbehavior,(b)detailofdiagramwithchaotic

oscillations.

thesetwoparametersaffectthecontrolleddynamicsoftheCDIreactioninessentiallythesamefashion.Discussion

BurstingandchaosemergeintheCDIreactionasaresultofafeedbackcontrolmechanism,whichregulatestheflowrateand/ortheratioofinputreagents.WeattributethisemergencetothespecificdynamicfeaturesoftheCDIreaction.Asmentionedabove,theCDIreactioninaCSTRwithoutfeedbackcontrolexhibitstwodifferentsteadystates,LIandHI,bistabilityofLIandHI,period-oneoscillations,andcoexistenceofperiod-oneoscillationswithHI(seeFigure9).AdynamicalphasediagramsimilartoFigure9ahasbeenobtainedfromexperi-mentaldataontheCDIreaction.16WesuggestthatthecoexistenceofoscillationswiththeHIsteadystateisresponsibleforburstinginthecontrolledsystem.

WeillustratetheemergenceofburstinginsetupsA,B,andCinFigure9a.Weassumethattheoscillatorystateofthesystem,justbeforethecontrolisimposed,ischaracterizedbypointSandthatthetargetconcentrationITissettobehigherthantheaverageconcentrationduringanoscillatorycyclebutlowerthaninanyHIsteadystate.Figure9ashowshowthestateofthesystemchangesduringonecycleofburstinginsetupA.TheoverallincreaseoftheflowratefromtheinitialpointSisillustratedbythearrowmarkedA+.BecauseITishigherthan[I-],theflowrateincreasesuntilitsvalueisequaltothatatthesaddle-node(limit)point.Duringthisincreaseink0,theoscillationsremainstable.WhenthevalueofthedynamicflowrateexceedstheHopfandlimitpointvalues,theoscillationsbecomeunstableandthesystemmonotonicallyapproachesthe

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