Early warning detection of runaway initiation using non—linear approaches

CommunicationsinNonlinearScience

andNumericalSimulation10(2005)

299–311

http://wendang.chazidian.com/locate/cnsns

Earlywarningdetectionofrunawayinitiationusing

non-linearapproaches

Jos??e-ManuelZald???var

aa,*,JordiBoschb,1,FernandaStrozzib,1,JosephP.Zbilutc,2

cInstituteforEnvironmentandSustainability,EuropeanCommission,JointResearchCenter,TP280,21020Ispra,VA,ItalybQuantitativeMethodsDepartment,CarloCattaneoUniversity,21053Castellanza,ItalyDepartmentofMolecularBiophysicsandPhysiology,RushUniversity,1653W.Congress,Chicago,IL60612,USA

Received7April2003;receivedinrevisedform8August2003;accepted8August2003

Availableonline4September2003

Abstract

Wepresentamethodforearlydetectionofrunawayinitiationinchemicalreactorsusingonlytem-peraturemeasurementsandbasedonthecalculationofthedivergenceofthesystem.Themethodisbasedonstatespacereconstructiontechniquesandisillustratedusingsimulatedaswellasexperimentaldatasets.Theresultsshowthatthemethodisabletodistinguishbetweenrunawayandnon-runawaysituationsanditdoesnotproducefalsealarmsduringcontrolledheating/coolingexperiments.

Ó2003ElsevierB.V.Allrightsreserved.

PACS:05.45;82.40.B;07.05.T;07.05.K

Keywords:Runaway;Reactorsafety;On-linedetection

1.Introduction

Thereliabilityofachemicalreactorinstalledinaplantdependsonthecapabilityofthecontrol/supervisionsystemtoestimateitsstateandtoidentify,intime,itsoperationmalfunctionsorfailuremodes.Speci?callyforchemicalreactorscarryingoutexothermicreactions,themajor

Correspondingauthor.Tel.:+39-332-789202;fax:+39-332-789328.

E-mailaddresses:jose.zaldivar-comenges@jrc.it(J.-M.Zald???var),fstrozzi@liuc.it(F.Strozzi),joseph_p_zbi-lut@http://wendang.chazidian.com(J.P.Zbilut).1Tel.:+39-331-572364;fax:+39-331-480476.

2Tel.:+1-312-9426008;fax:+1-312-9428711.

1007-5704/$-seefrontmatterÓ2003ElsevierB.V.Allrightsreserved.

doi:10.1016/j.cnsns.2003.08.001*

300J.-M.Zald???varetal./CommunicationsinNonlinearScienceandNumericalSimulation10(2005)299–311

problemisthelossoftemperaturecontrol.Inthissituation,whentherateofheatgenerationbychemicalreactionexceedstherateofheatremovalbythecoolingsystem,thereisapositivefeedbackmechanism,sincethetemperatureofthereactionmasswillrise,increasinginturntheheatgenerationrate.Inthissituation,ifnocountermeasuresaretaken,arunawayorthermalexcursionmayoccur.

Thereareseveralreasonsforwhichtheearlydetectionofdangerousstatesisimportantforthecorrectoperationofchemicalreactors.Apartfromhealthandenvironmentalconsiderations,anaccidentinachemicalreactorisextremelyexpensive[7].Furthermore,chemicalindustriesex-perienceanumberofnear-misses,whichcouldbereducedbytheuseofanearlywarningdetectionsystem(EWDS).Thisreductionwouldalsocontributetothereductionofminorandmajorac-cidents,sinceasithasbeenshownthatthesevaluesarecorrelated[12].

Methodsforearlywarningdetectioncanbedividedintothreemaincategoriesdependinguponthequantitiesbeingused:conventionallimitchecksystems[10,16],temperaturederivatives[8],andmodel-basedestimationtechniques,i.e.Kalman?lteringorequivalent[6,9].However,despiteallthedi?erentapproaches,theearlyon-linedetectionofhazardousstatesinbatchandsemibatchprocessesisstillanopenproblembecauseofthewiderangeofprocessesthatarecarriedoutonsuchequipment;theircomplexity;strongnon-linearities;andtime-varyingparameters.Inthiscontext,thebottlenecksforthedevelopmentofane?ectiveon-linedetectionsystemarethede?nitionofcriteria,whichdistinguishbetweendangerousandnon-dangeroussituations;theavoidanceoffalsealarms,sincecountermeasuresmayconsistondumpingthereactorcontentsorinjectinganinhibitorwiththelossofthebatch,whichmeansthatinpracticeatrade-o?existsbetweenearlydetection(sensitivity)andnumberoffalsealarms(reliability);andtheuseoffewmeasurementssincethenumberofprocesscarriedoutbatchwisemakesitnoteconomicallyfeasibletodevelopadetailedkineticmodelforeachprocess,i.e.theEWDShastobeasinde-pendentaspossibleoftheactualprocesscarriedoutintheplant.

Furthermore,asshowninFig.1thedynamicbehaviourofchemicalreactorshaveahighparametricsensitivity[17],i.e.asmallchangeinoneoftheoperatingconditionsorparameters

ofFig.1.Reactortemperaturesmeasurementsrecordedduringseveralsemibatchprocessesusingthesystem2-butanol–propionicanhydrideandsulphuricacidascatalystina2lbenchscalereactor.Alltheexperimentswereperformedinsimilaroperatingconditionsbutchangingin(a)thejackettemperature,from10to80°C;andin(b)thecatalystconcentration,from0.0to2.5wt.%.

J.-M.Zald???varetal./CommunicationsinNonlinearScienceandNumericalSimulation10(2005)299–311301thesystemmayproduceanenormouschangeinitsdynamicbehaviour.Forexample,inFig.1bachangeofsulphuricacidconcentrationof0.2wt.%isenoughtopassfromthenon-ignitionregiontoarunawaysituation.Ofcourse,whenoperatingchemicalreactorsonetriestoavoidworkinginsuchregions,sinceaprocesschange––letssayindustrialwatertemperature,fromsummertowinter––couldproducecompletelydi?erentbehaviourintheprocess,leadinginsomecasestoarunawayevent.

Inaseriesofrecentworks[14,19],anewcriteriontodelimitrunawayboundarieswasdevelopedapplyingtechniquesfromnon-lineardynamicalsystemstheorytocharacterizethesensitivityofchemicalreactors.Therunawaydetectioncriterionwasde?nedaswhenthedivergenceofthereactorbecomespositiveonasegmentofthereactionpath,i.e.div>0.Werecallthatthedi-vergenceisascalarquantityde?nedateachpointasthesumofthepartialderivativesofthemassandenergybalanceswithrelationtothecorrespondentvariables––temperatureandconversions.Inthiswork,wehaveanalyzedtheon-lineapplicationofthedivergencecriterionfortheearlywarningdetectionofrunawayinisothermalbatchreactors,i.e.chemicalreactorsinwhichatemperaturecontrolsystemisinplace.Normally,temperaturecontrolisachievedbyadjustingthetemperatureofthethermo-vector?uidcirculatingthroughajacketthatsurroundsthereactorand,hence,thedynamicbehaviouroftheheating/coolingcircuitswillin?uencethedynamicsofthereactortemperatureaswellasitssafety.Inthiscase,itisnecessaryconsiderforstatespacereconstructionnotonlythereactortemperature[2],butalsothejackettemperature.Oncethedynamicshasbeenreconstructed,thedivergencecanbecalculated.Wehave?rstvalidatedtheresultsusingsimulateddata.Thishasallowedthedeterminationofthemostsuitablerecon-structionstrategy.Finally,theexperimentalvalidationhasbeencarriedoutusingdatafromabench-scale2lreactorandfroma100lpilotplantreactor.Theresultsshowthatthesystemisabletodistinguishbetweenrunawayandnon-runawaysituationsanditdoesnotproducefalsealarmsduringheating/coolingduetochangesinthereactortemperatureset-point.

2.Experimental

2.1.Statespacereconstruction

Inordertocalculateon-linethedivergencewithouttheneedofknowingthedi?erentialequationsofthesystemwehaveusedthetheoryofembedding.Thetheoryofembeddingisawaytomovefromatemporaltimeseriesofmeasurementstoastatespacesimilar––inatopologicalsense––tothatoftheunderlyingdynamicalsystemweareinterestedinanalysing.TechniquesofstatespacereconstructionwereintroducedbyPackardetal.[11]andTakens[15],whoshoweditispossibletoaddressthisproblemusingmeasurementsofatimeseriesofthedynamicalsystemofinterest.Inageneralcase,letsðtÞbethemeasureofsomevariableofoursystem,whichisrelatedtothestatevariablesbyanunknownfunctionh,

sðtÞ¼hðxðtÞÞð1ÞTakens[15]provedthat,undercertainconditions,thedynamicsontheattractoroftheunderlyingoriginalsystemhasaone-to-onecorrespondencewithmeasurementsofalimitednumberofvariables.Eventhough,theoretically,allthemethodsofreconstructionshouldproducethesame

302J.-M.Zald???varetal./CommunicationsinNonlinearScienceandNumericalSimulation10(2005)299–311

result,thereisnoapriorimethodtodecidewhichisthebest[4]andthereisaconsiderabledi?erenceinthequalityoftheresultingreconstructedstatespacecoordinatesandhenceinthecalculatedquantitiesfromthesecoordinates.Thelackofauniquesolutionforallpurposesisdueinparttothepresenceofnoiseandthe?nitelengthofthedataset[3].Furthermore,inourcasethesituationiscomplicatedbythenon-stationarityofoursystem.

Inthisworkwehavetestedseveralmethods:timedelayembeddingvectors{sðtÞ;sðtÀDtÞ;sðtÀ2DtÞ;...;sðtÀðdEÀ1ÞDtÞ};derivativecoordinates{sðtÞ;dsðtÞ=dt;...;dðdEÀ1ÞsðtÞ=dtðdEÀ1Þ}andintegralcoordinates{sðtÞ;I1½sðtÞ??;...;IdEÀ1½sðtÞ??}.Astheresultsweresim-ilarwedecidedtousederivativecoordinates.Theadvantageofderivativecoordinatesistheirclearphysicalmeaning;theirdrawbackliesintheirsensitivitytonoise.Inthiscaseweneedtoestablishonlyoneembeddingparameter,i.e.theembeddingdimension,dE.Theembeddingdimensionisthedimensionofthestatespacerequiredtounfoldthesystemfromtheobservationofscalarsignals.

2.2.Divergencecalculation

Oncethestatespacehasbeenreconstructed,inourcasefromtemperaturemeasurements,itispossibletocalculatethedivergence,div,ofthesystem,thatshouldbepreserved[1].Thereareseveralmethodstocalculatethedivergencewhicharerelatedtothevolumeofthesysteminphasespace(ps)byLiouvilleÕstheorem[1]whichstatesthat??Zt divfF½xðsÞ??gdsð2ÞVpsðtÞ¼Vpsð0ÞÁexp

where

divfF½xðtÞ??g¼oF1½xðtÞ??oF2½xðtÞ??oFd½xðtÞ??þþÁÁÁþox1ox2oxdð3Þ

Afterseveralmanipulationsitispossibletoarriveatthefollowingexpression:

_psðtÞVdiv¼VpsðtÞð4Þ

Duetothevolumecontractioninstatespace,characteristicofdissipativesystems,VpsðtÞwillrapidlytendtozeroandproduceartefactswhenintroducedasthedenominatorinEq.(4).However,byde?nitionthevolumeisalwayspositiveandhence,div>0isequivalenttocheckingif

DVpsðtÞ>0ð5ÞwhereDVpsðtÞisanin?nitesimalstatespacevolumevariation.Thiseliminatestheneedtodividethetwosmallnumbersthatproducesanincreaseinthenumericalerrorsofthecalculation.EventhoughtheabsolutevaluesofDVpsðtÞarenotpreservedunderstatespacereconstruction,itssign,whichistheidenti?cationcriteria,willbepreserved.

Severalmethodstocalculatethedivergenceusingoneormoretemperaturetrajectories(measurements)havebeendeveloped[2].Herewereportonthecalculationusingonlyonetemperaturesensorinsidethereactorandoneinthejacket,withanembeddingdimensionofthree

J.-M.Zald???varetal./CommunicationsinNonlinearScienceandNumericalSimulation10(2005)299–311303

and{T;dT=dt;dðTÀTwÞ=dt}asstatespacevariablesforthereconstruction.Fig.2showshowtoreconstructthedivergenceusingonlyonetemperaturemeasurementinsidethereactor,assumingthattheJacobianremainsconstant.Inthiscasethetimebetweentwovolumecalculationsisde?nedasDt1,whereasthetimebetweentwostatespacepoints,Q0andQ1,isde?nedasDt2:Inthex-axisthereactortemperature,T,isrepresented,whereasinthey-axisandz-axisthe?rstderiv-ativesofreactortemperature,dT=dt,andthedi?erencebetweenreactorandjackettemperatures,dðTÀTwÞ=dt,arerepresented.

Inthiscasethevolume(dE¼3)iscalculatedas??2x3??xxxxx????QÀQQÀQQÀQ102030????yyyyy5??4QyÀQQÀQQÀQð6ÞdetVpsðtÞ¼??102030????zzz????QzQzQz1ÀQ02ÀQ03ÀQ0

wherej...jstandsforabsolutevalueanddetreferstothedeterminantofthematrix.Then

DVps¼VpsðtÞÀVpsðtÀDt1Þ

Dt2ð7Þ

2.3.Simulatedprocesses

Inordertoavoidproblemsrelatedtonoiseinthetemperaturemeasurements,wedecided,asa?rststep,tousethesimulatedtemperaturedataobtainedforanisothermalbatchreactorinwhichnoreaction(heating/coolingpro?les)ora?rstorderreactiontakeplace.Thiswillallowustochoose,bycomparingthetheoreticalandcalculateddivergence,themostappropriate,i.e.best?tand/orleastnumberfalsealarms,reconstructionschemeinabsenceofnoise.Thedi?erential