Reactorrunawaycriteriacanbeappliedtodefinetheboundaries ofsafeandunsaferegimesthroughdistinguishingtherunawayand non-runawaystates. Thisfeatureallowstoapplycriteriain off-linetasks(likeprocessdesign,optimization)andinon-linetasks too(likeearlywarning).Therefore,thermalrunawaycriteriaare applicableindesigningandoperationofchemicalreactors(Jiang etal.,2011).Abriefhistoryaboutthereactorrunawaycriteriauntil 2006canbefoundin(Shouman,2006).
Thermalrunaway criteria can be classifiedinto three types, which are geometry-based criteria, stability-based criteria and sensitivity-based analysis can beperformed to define runaway boundaries,whicharepresentedinthefollowingSections5.2–5.4.
Therunawaycriteriaandtheyearoftheirfirstpublicationare pre-sentedinTable3.Section5.1presentsasimplemathematicalmodel ofatubularreactor(orbatchreactor),onwhichthederivationof runawaycriteriacanbepracticedeasily.
5.1. Mathematicalmodel
Afirstorderreactioncarriedoutinabatchreactorispresentedin thissectionwhichwillprovideasabaseforpresentationofthermal runawaycriteria.Thereactorwasconsideredasperfectlymixedso
Table3
Thermalrunawaycriteriadevelopmentsovertime.
Criterion Yearofpublication Reference
Semenov-criterion 1928 Semenoff(1928),
Semenov(1940)
“PracticalDesign”criterion 1938 Berty(1999)
vanHeerdencriterion 1953 vanHeerden(1953)
Gilless-Hoffmanncriterion 1961 Berty(1999),Gillesand Hofmann(1961) ThomasandBowescriterion 1961 Thomas(1961),Varma
etal.(2005) AdlerandEnigcriterion 1964 AdlerandEnig(1964) vanWelsenaereandFroment
criterion
1970 vanWelsenaereand
Froment(1970) Morbidelli-Varmacriterion 1987 MorbidelliandVarma
(1988)
Adiabaticcriterion 1988 Gygax(1988)
Hopf-bifurcationanalysis 1989 Colantonioetal.(1989)
Vajda-Rabitzcriterion 1992 VajdaandRabitz
(1992)
Strozzi-Zaldivarcriterion 2003 Zaldívaretal.(2003)
Lyapunov-stability 2006 Szeifertetal.(2006)
Adiabaticcriterionbasedon Strozzi-Zaldivarcriterion
2016 Guoetal.(2016)
Kähm-Vassiliadiscriterion 2018 KähmandVassiliadis (2018a)
ModifiedSlopeCondition 2019 KummerandVarga
(2019a)
ModifiedDynamicCondition 2019 KummerandVarga (2019a)
thefollowingdifferentialequationscanbewrittentodescribethe dynamicalbehaviour: Fig.5showshowthepresentedmodel(Eqs.6–11)issensitiveto
thewalltemperature,anditpresentsthedevelopmentofthermal runaway.
5.2. Stability-basedcriteria
Thestateofthesystemcanbeconsideredstableifafterasmall disturbancethesystemreturnstoinitialstateandduringthe tran-sientbehaviourthestateofthereactorstaysclosetothatinitial state.Thistheorycanbeusedtoinvestigatereactorrunawaysince incaseofrunawayreactionssimilarsituationoccurs,wherethe positivefeedbackinthetemperatureandreactionraterelationship canresultinthedevelopmentofrunaway.Thatfirststateofthe sys-tem,whenrunawayisoccurredcanbeconsideredasunstablestate, fromwhichthereactorcannotgobacktotheinitialstate. Numer-ousstability-basedrunawaycriteriawereproposedtoindicatethe developmentofthermalrunaway,whicharenowpresentedinthe followingsection.
5.2.1. Semenov-criterion
Firstpioneerworkinthefieldofreactorrunawaywasdoneby Semenov,whichworklaidthegroundworkforfurtherresearches.
Thissectioniswrittenbasedon(Stoessel,2008;Semenoff,1928;
Semenov,1940).Semenovconsideredanexothermalreactionwith zero-orderkinetics.Semenov-diagrampresentstheheat-releasein reactionand theremovedheatbyheattransferasafunctionof temperature.
Fig.7 presents therelationship betweenthe generated and removedheat,wherethegeneratedheatvariesexponentiallywith processtemperature,whiletheremovedheatvarieslinearlywithit.
ThreeessentialpointsdrawattentioninSemenov-diagram,which aremarked asA, Band C, andthe belongingtemperaturesare markedasTw1, Tw2 andTw3.InAwecanrespectastableoperating pointsinceifthecoolingtemperatureislowerthan Tw2,the pro-cesstemperaturewilldecreaseduetothehigherremovedheat untilA,andnoself-ignitionoccurs.Ifthecoolingtemperatureis higherthan Tw2,self-ignitionoccurssince thegenerated heatis continuouslyhigherthantheremovedheat.Cpointrepresentsthe criticalpointincaseofahighercoolingtemperature,wherethe generatedheatcurveistangentatonepointtotheremovedheat line.Thebelongingcoolingtemperatureisconsideredascritical, orasthelowesttemperatureofself-ignition.Inthispointalittle increaseincoolingagenttemperaturethecoolinglinewillhaveno intersectionbetweenthegeneratedheatandremovedheatcurve leadstotherunawayofreaction.
Fortheaimofavoidingthermalrunawayitisnecessaryto oper-atethereactor faraway from critical conditions. Based onthe Semenov-diagramandfurtherinvestigationofthecriticalpointa runawaycriterioncanbederived.Inthecriticalpointthegenerated andremovedheat,andalsotheirderivativeswithrespectto tem-peratureequals,thiscanbewrittenasEqs.12–15presents.Since thereagentconsumptionisneglected,thereactionratevariesonly withtemperature,hencethepartialderivativeofthereactionrate canbeconsidered.
Dividingthe13and15.equationsthefollowingcriticalequation istheresult:
r rT =RTc2
E =(Tc−Tw)=Tc (16)
Eq.16 presents that thereis a minimal temperaturedifference betweentheprocessandcoolingtemperaturetokeepthe reac-tionoperationstable.Semenov-diagramhelpsustoformulatethe runawaycriterion,becausethecriticaltemperaturedifferenceis alwayssatisfiedwhenthetemperatureisbelowthecritical tem-peraturevalue.
(T−Tw)≤ RTc2
E (17)
FromEq.17thecriticaltemperaturecanbecalculatedbysolving thequadraticequation. Ifweconsideronlythefirsttwotermsontherightside,the followingrunawaycriterion(Semenov-criterion)canbederived:
(T−Tw)≤ RTw2
E (19)
WepaytributetotheSemenov-number,whichistheratioof dimensionlessreactionheatparameterand theheattransfer,as follows:
= (−Hr)kcn UA
E
RT2 (20)
Forverylargeactivationenergiesthefollowingcriterioncanbe defined,mentionedintheliteratureasSemenov-criterion(where eisthenaturalnumber):
<1
e= c (21)
Thisequationisdeterminingintheresearchfieldofthermal ignition,becausethefollowingresearchesfocusonhowto deter-mine thecritical Semenov-number in more realistic cases, like withoutneglectingthereactantconsumption.
However,wearegoingtopresenttherunawaycriteria with-outinvestigatingtheconcretevalueofSemenov-numbersinthe followingsections,insteadwearegoingtopresentthebase the-ory.Criticalstates(temperature,concentration,etc.)canbedefined though,andthecriticalSemenov-numberscanbecalculatedfrom thesevariables.
5.2.2. VanHeerdenand“practicaldesign”criterion
BertyclearlypresentedthetheorybehindVanHeerden crite-rion,whichisoftencalledas“SlopeCondition”(Berty,1999;van Heerden, 1953). In a steady-state operationthe generated and removedheatareequal. Itisevidentalsothattheheat genera-tionand heatremovalrateincreaseswithtemperature,butthe generatedheatincreasesexponentially.Ifthereisanydisturbance inthereactortemperaturetheheatremovalrateshouldincrease fasterwithtemperaturethanthegeneratedheat,itwouldprevent temperaturerunaways.Mathematicalformofthecriterionisthe following:
dqgen
dT ≤dqrem
dT (22)
TheareaofsensitivedomainwasdefinedbyVanHeerdenin 1953(vanHeerden,1953).Perkinsassumedzeroorderkineticsto defineasafeboundary.ConsideringEqs.22and12thefollowing criterioncanbedefined:
T−Tw≤RT2
E (23)
Bashiretal.derivedthesamecriterioninvestigatingthe inflec-tionpointinageometricplane(Bashiretal.,1992),statingthatthe calculatedmaximumtemperatureinEq.23isthelimitingvaluefor runawayattheinflectionpoint.
5.2.3. Gilles-Hoffmanncriterion
Gilles and Hoffmannin1961 recognizedthe“Dynamic Con-dition”, whichistheconditionthat setsthelimitstoavoidrate oscillation.Criterionisstatedastheincreaseofheatremovalrate withtheincreaseoftemperaturemustbelargerthanthe differ-encebetweenheatgenerationrateincreaseduetotemperature alone andreaction ratedecrease duetotheconcentrationdrop alone(Berty,1999;GillesandHofmann,1961).
∂qgen
∂T
c
+ ∂m
∂c
T
≤dqrem
dT (24)
wheremisthematerialbalancefunction.
5.2.4. Lyapunov-stabilityingeometric-andphase-plane
Szeifertetal.proposedtouseLyapunov’sindirectmethodto forecastreactorrunaway(Szeifertetal.,2006;Sastry,1999).The
stabilityanalysisofasystemdefinedbyasetofnonlinear differ-entialequationsofthestatevariablesapplyingLyapunov’sindirect methodisreducedtoaneigenvalueanalysisoftheJacobianmatrix.
J= ∂f
∂x (25)
IfrealpartofeacheigenvaluesoftheJacobianmatrixisnegative thenthemodelisstable,butifanyofthesearepositivethensystem isunstableattheinvestigatedoperatingpoint.Lyapunov-stability canbeperformedingeometric-andinphase-planetoo.Thespatial stabilitycriterionisalwaysmoreconservative,becausethe stabil-ityinphasespacealwaysfollowsfromthespatialstabilitywhile inverselydoesnot.
In2008López-Garcíaetal.proposedtoinvestigatethe steady-statesolutionswithaperturbationmodel,becausethedynamic studyisessentialtoguaranteethethermallystableoperation.The methodis basedonthelinearization oftheperturbationmodel whichresultintheanalysisoftheeigenvaluesofJacobianmatrix (López-GarcíaandSchweitzer,2008).VajdaandRabitzsimilarly investigatedtheperturbationmodelearlierin1992,butthey inves-tigatedthesensitivityofmaximumvalues ofeigenvaluesofthe Jacobianmatrix(VajdaandRabitz,1992).
Forinvestigating thedynamicsof asystem,Hopf-bifurcation analysiswassuggested,whichisbasedoninvestigatingthe eigen-values too.Ifthe realpartof a complex-conjugate pairs ofthe Jacobianmatrixbecomespositivethenbifurcationoccurs,andthat meansreactorrunawaymaydevelop(Colantonioetal.,1989;Ball andGray,2013;GómezGarcíaetal.,2016;McAuleyetal.,1995;
Kimetal.,1991;BallandGray,1995;Ball,2011).
5.2.5. Strozzi-Zaldivarcriterion(Divergencecriterion)
StrozziandZaldivarinvestigatedthephase-spacevolume con-tractionsduringthereactoroperationbasedoninvestigatingthe Lyapunov-exponentsand thedivergence ofthe system(Strozzi etal.,1999).Ithasbeenshownthatthedivergencecriterioncan beappliedfordevelopingsafetyboundarydiagramstodistinguish therunawayandnon-runawaystatesforseveraltypesofreactors (BR,SBR,CSTR)andformultiplereactions,alsowithandwithout ofacontrolsystem(Zaldívaretal.,2003).
StrozziandZaldivarprovidedthefollowingderivationoftheir runawaycriterion(Strozzietal.,1999).AccordingtotheLiouville’s theorem,contractionofastatespacevolumeofad-dimensional dynamicalsystemcanbedefinedbasedonitsdivergence(Arnold, 2006).
dV(t) dt =
divF[x(t)]dx1(t)...dxd(t) (26) wherethedivergenceofthesystemcanbecalculatedas
divF[x(t)]=∂F1[x(t)]
∂x1(t) +∂F2[x(t)]
∂x2(t) +···+∂Fd[x(t)]
∂xd(t) (27) Assumingthatthed-dimensionalvolumeissmallenoughthat thedivergenceofthevectorfieldisconstantoverV(t),then dV(t)
dt =V(t)divF[x(t)] (28)
IntegratingEq.28theinitialphase-spacevolumeV(0)changeswith timeas
V(t)=V(0)exp
t 0
divF[x(t)]d
(29) Hencetherateofchangeofthestate-spacevolumeisgivenby thedivergenceofthesystem,whichislocallyequivalenttothetrace oftheJacobianofF.Theexpansionandcontractionofthe state-spacevolume,sothatthedivergenceoftheinvestigatedsystemis inrelationwithrunawayandnon-runawaysituations.Practically
itmeansthatifthestatevariablesdriftoffforasmall perturba-tionthenthesystemisunstable.Incasethedivergenceisnegative therewillbenorunaway,althoughifthedivergenceispositive, runawaywilldevelop.Therefore,theproposedrunawaycriterion isthefollowing:
divF[x(t)]≤0 (30)
Copelli etal. modified theoriginal divergence criterion,and theyproposedtodisregardallcontributionsarisingfrom extent-of-reactions that are not related toheat evolution. Other state variablescangenerateastrongstate-spacevolumecontractionthat isnotrelatedtothedevelopmentofrunawaywhichmayleadsto thefailureofdivergencecriterioninpredictingreactorrunaway.It meansthatforexamplethecomponentswhicharenotreactantare neglectedwhenevaluatingthemodifieddivergenceofthesystem (Copellietal.,2014),(Kahm,2019).
Strozzietal.alsoinvestigatedtheLyapunov-exponentstodefine sensitivity.Lyapunov-exponentcanmonitorthebehaviouroftwo neighbouringpointsofasysteminadirectionofthephasespaceas afunctionoftime:IftheLyapunov-exponentispositive,thenthe pointsdivergefromeachother,iftheexponentbecomesnegative, thenthepointsconverge.Lyapunov-exponentsarerelatedtothe eigenvaluesoftheJacobianmatrix,sinceitaveragestherealparts ofalleigenvaluesalongatrajectory(Strozzietal.,1994;Strozzi andZaldívar,1994).AlthoughtheLyapunov-exponentscan under-estimatetherunaway boundaryfor likeautocatalyticreactions, becauseitusestheintegralovertimewhich isslowtorespond tofastchange.Therefore,Strozzietal.proposedtoapply diver-gencecriterion(Strozzietal.,1999).Kähmetal.laterinvestigated theLyapunov-exponentsnotinsensitivitycontext,but investigat-ingthevaluesofit.IftheLyapunov-exponentbecomespositive,an unstableprocessispresent(KähmandVassiliadis,2018a;Kähm andVassiliadis,2018b;KähmandVassiliadis,2018c).
We cancalculate thedivergenceonline, withoutneeding to knowthedifferentialequationsofthesystembyusingthetheory ofembedding.Statespacereconstructionisapossibletechnique toaddressthis problemusingtimedelayembeddingvectorsof theoriginalmeasurements(i.e.,temperatureorpressure measure-ments)(Boschetal.,2004a;Boschetal.,2004b).Althoughthereis severalmethodsofreconstruction,butthereisnoapriorimethodto decidewhichoneisthebest.In(Zalı ´dvaretal.,2005)Zaldivaretal.
testedseveralmethods:timedelayembeddingvectors;derivative coordinatesandintegralcoordinates,buttheresultsweresimilar andtheyusedderivativecoordinatesbecauseoftheirclearphysical meaning.Therearetworeconstructionparameters:theembedding dimension,andthetimedelay.Theembeddingdimensionisthe dimensionofthestatespacerequiredtounfoldthesystemfrom theobservationofscalarsignals,whereasthetimedelayisthelag betweendatapointsinthestatespacereconstruction(Boschetal., 2004b).
Guoetal.developedanadiabaticcriterionbasedonthe diver-genceofanadiabaticmodelofthereactorsystemwithzerofeed rateresultinamorestrictrunawaycriterion(Guoetal.,2016;Guo etal.,2017a).
WalterKähmdevelopedastabilitycriterionbasedonthe origi-naldivergencecriterion,whichisbasedonthedifferencebetween thedivergenceoftheJacobianmatrixoftheinvestigatedreactor systemvariablesandthecorrectionfunction.Thecorrection func-tionisderivedasafunctionofthedivergenceoftheJacobianatthe previoustimestep;Damköhlernumber;Barkelewnumber; Arrhe-niusnumberandtheStantonnumber.Theyintroducedthisstability criterion,becausedivergencecriterionmayoverpredictthe ther-malrunawaypotentialofthesystem.Thederivationisbasedon a linearapproximationofthedivergence(KähmandVassiliadis, 2018a;Kahm,2019;KähmandVassiliadis,2018d).Theproposed
stabilitycriterionissuccessfullygeneralizedformultiplereactions (KähmandVassiliadis,2019).
5.2.6. Modifieddynamicandslopecondition
KummerandVargainvestigatedthemostfrequentlyapplied criteriaand derived two newcriteria as a result(Kummer and Varga,2019a).Eq.31presentstheModifiedSlopeCondition(MSC) andEq.32presentstheModifiedDynamicCondition(MDC).We investigatedthreedifferentreactionsystems(singlereactionwith areagent,twoparallelreactions,andanautocatalyticreaction sys-tem)tovalidatetheModifiedDynamicandSlopeConditioncriteria, whichinthereliabilityandthetimeofindicationwerecompared.
MDCdidnotmissanythermalrunawaydevelopment,butthe per-formanceofMSCiscompatiblewiththeinvestigatedones.
∂qgen
Several reactor runaway criteria exist based on a geomet-ric characterization of temperature trajectories, which will be presentedinthis section.Advantagesof inflexion-basedcriteria (ThomasandBowes-,AdlerandEnigcriterion)andadiabatic cri-terionisthatitrequiresonlyatemperatureprofileortrajectory toevaluatethereactionstates,althoughwithoutinvestigatingthe states on a prediction horizon the runaway indications proba-blyoccurslately.Inflection-basedcriteriadonotgiveinformation abouttheintensityofthereactorrunaway.VanWelsenaereand Fromentcriterionisquiteconservativethoughandindicates reac-torrunaway quite early,but a model of thereactor system is requiredfortheapplication.
5.3.1. ThomasandBowescriterion
ThomasandBowesproposedtoindicatereactorrunawayasthe situationinwhichaninflexionpointappearsbeforethe tempera-turemaximuminthegeometricplane(inversustimeorlength).It meansthatthereactoroperationstayscontrollableifthefollowing statementsaresatisfied(Thomas,1961;Varmaetal.,2005).
d2T
dt2 <0whiledT
dt >0 (33)
DenteandCollinain1964independentlyproposedthesame criterion(Varmaetal.,2005).
5.3.2. AdlerandEnigcriterion
AdlerandEnigfounditmoreconvenienttoworkina phase-plane(intemperature-conversion)thaninthegeometricplane.To indicatereactorrunaway aninflexionpointmustappearbefore thetemperaturemaximuminthephase-plane.Itmeansthatthe reactoroperationstayscontrollableifthefollowingstatementsare satisfied,wherexistheconversion(AdlerandEnig,1964).
d2T
dx2 <0whiledT
dx >0 (34)
5.3.3. VanWelsenaereandFromentcriterion(orMaxicriterion) vanWelsenaere and Fromentdetermined critical conditions basedonthelocusoftemperaturemaximainthe temperature-conversion plane. This criterion can be eliminated based on obtainingtherelationbetweenmaximumprocesstemperatures
Fig.6. Sensitivityofthereactormodelwithrespecttowalltemperature.
Fig.7.Semenov-diagram.
evolvingatdifferentcoolingagenttemperatures(vanWelsenaere andFroment,1970).
dT
dx >0dcm
dTm
>0 (35)
5.3.4. Adiabaticcriterion
Afrequentlyappliedrunawaycriterion(eveninindustrial appli-cation)isthattheprocesstemperatureevolvingunderadiabatic conditions(sotheMTSR)cannotexceedtheMaximumAllowable Temperature(Abeletal.,2000).
Tp+Tad=MTSR≤MAT (36)
5.4. Sensitivityanalysisofchemicalreactors(Morbidelli-Varma criterion)
A.Varmaetal.wroteanexcellentbookabouttheparametric sensitivitiesinchemicalsystems(Varmaetal.,2005).Theanalysis ofhowasystemrespondstochangesintheparametersiscalled parametricsensitivity(Varmaetal.,2005).Inthecontextof chemi-calreactorsBilousandAmundsonperformedapioneerworkonthe fieldofparametricsensitivity,wheretheresearchersshowedhow themaximumtemperaturealongthereactorlengthvarieswith theambient(cooling)temperature(BilousandAmundson,1955;
BilousandAmundson,1956;Grayetal.,1981;Emigetal.,1980;
Grayetal.,1981).Theresultofasimilaranalysiscanbeseenin Fig.6.Sensitiveregionsofoperationsshouldbeavoidedbecauseits
performancebecomesunreliableandchangessharplywithsmall variationsinparameters.Althoughsomeexperimentalstudiesare availableintheliterature(Emigetal.,1980;LewisandVonElbe, 2014),itisdifficulttoperformwholesomeinvestigationsaboutthe reactionsystems(nottomentiontheindustrialsystems),because thesesystemsinvolvemany parametersaffectingthebehaviour ofthereactor.Therefore, modelbasedinvestigations are neces-sary.For theaimofinvestigationthesensitivity ofreactors we shoulddefinevaluableoutputs(dependentvariables),and valu-ableinputs(independentvariables).Dependentvariablescanbe investigatedingeometric-or/andinphase-plane,whichcanbefor exampleproductivity,processtemperature,processpressureetc.
Inputvariablestypicallyareinitialconditions,operatingconditions andgeometricparametersofthesystem.
MorbidelliandVarmausedthefactthatneartheexplosion (run-away)boundarythesystembehaviourbecomessensitivetosmall changesinsomeoftheinputorinitialparameters,andtheydefined theboundarybetweenrunawayandnon-runawayzonebasedon thissensitivityconcept.Thefirst-orderlocalsensitivityorabsolute
MorbidelliandVarmausedthefactthatneartheexplosion (run-away)boundarythesystembehaviourbecomessensitivetosmall changesinsomeoftheinputorinitialparameters,andtheydefined theboundarybetweenrunawayandnon-runawayzonebasedon thissensitivityconcept.Thefirst-orderlocalsensitivityorabsolute