NIPPONSTEELTECHNICALREPORTNo.89January2004-46-UDC621.746.27:681.3*1Environment&ProcessTechnologyCenter,TechnicalDevelopmentBureauFormulationofMoldLevelControlModelbyMoltenSteelFlowAnalysisMethodDaiSUZUKI*1AbstractThispaperdescribesactivitiesrelatingtomoldlevelcontrolincontinuouscastingprocessascaseexamplesofourprocesscontrolsolutionbusiness,inwhichweclarifyandevaluatetheprocessphenomenaanddesignthecontrolsystem.Initially,amoldlevelcontrolmodelhasbeenformulatedbyfluidflowanalysistechniquethatevaluatesmoltensteelflowinthemoldquantitatively.Themodelcalculatesthefluidflowandfreesurfaceinthemoldbymoltensteelflowanalysismethod,whilecontrolsystemanalysisisperformedsimultaneously.Itenablesrealisticsimulationofthemoldlevelcontrolsystemconsideringthefluidflowturbulence.Asaconse-quence,moldlevelcontrolinhigh-speedcastingcanbepredictedandevaluatedwithhighaccuracy.1.IntroductionMoldlevelcontrolisafunctionoftheprocesscontrolofcontinu-ouscastingtomaintainthelevelofmoltensteelsurfaceinamold(hereinafterreferredtoasthemoldsteellevel)constant.Thishasasignificantinfluenceoverthequalityandyieldofthefinalproduct.Thereisaclosecorrelationbetweenthefluctuationofthemoldsteellevelandtheoccurrenceofsurfacedefectsoffinalproducts:itisconsideredthat,whenthemoldsteellevelfluctuatessignificantly,castingpowderandotherimpuritiesfloatingonthesurfaceofmol-tensteelareentrappedinsteelandtheyappearintheformofsurfacedefectsofsteelsheetproductsduringrolling.Topreventthis,moldlevelcontrolisdesignedsoastominimizethefluctuationofthemoldsteellevel.Thedynamicsofthemoldsteellevelis,roughlyspeaking,asimpleintegralsystemandforthisreason,theeffectofmoldlevelcontrolhasconventionallybeencalculatedinitsdevelopmentstageonanassumptionofanintegralsystem.Inpractice,however,disturbancesinmoltensteelflowtendtoshowintheformofsurfaceripplesasthecastingspeedincreases.Asaresult,withaconventionalcrudemodel,itwasdifficulttoobtainareliablesimulationresultanditwasim-possibletoaccuratelypredictandevaluatethecontrolperformanceofamoldlevelcontrolsystemunderaconditionofahighcastingspeed.Inviewofthesituation,theauthorhasformulatedamoldlevelcontrolmodelapplyingmoltensteelflowanalysis.Fig.1showsablockdiagramofthedevelopedmodel.Thismodelischaracterizedbydescribingthemoltensteelflowinamoldandthedynamicsofamoltensteelsurfacefromtheviewpointoffluidflowanalysisandcombiningthemwithcontrolsystemanalysis.Asaresult,amoldPlantmodel(Moltensteelflowanalysis)LevelcontrolFreesurface(distributed)SensorControllerDetectedlevelPouringvelocity1.3-Dnumericalfluiddynamicsmethod2.ContinuityEq.,incompressibility3.Navier-StokesEq.4.Finitedifferencemethod5.Turbulencemodel:Largeeddysimulationmodel6.Freesurfaetracking:VolumeoffluidmethodFig.1MoldlevelcontrolmodelbasedonmoltensteelflowanalysisNIPPONSTEELTECHNICALREPORTNo.89January2004-47-levelcontrolsimulationinconsiderationoftheturbulenceofmoltensteelflowwasmadeviable,anditbecamepossibletoaccuratelypredictandevaluatetheperformanceofmoldlevelcontrolatahighcastingspeed.2.ContinuousCastingProcessBeforediscussingthemoldlevelcontrolmodel,continuouscast-ingequipmentandmoldlevelcontrol,whichconstitutetheback-groundofthemodel,arebrieflyexplained.2.1ContinuouscastingequipmentContinuouscastingequipmentaresteelproductionfacilitiesforefficientlyproducingslabs(orbloomsorbillets)forthesubsequentrollingprocessbycontinuouslysolidifyingmoltensteelafteritsre-fininginasteelmakingfurnacesuchasaconverter.Moltensteeldischargedfromaladleistemporarilystoredinatundishandthenpouredthroughanimmersionentrynozzleintoamoldtheinsidewallsofwhicharelinedwithwater-cooledcopperplates.Thesolidi-ficationofthecaststeelbeginsatitsinterfacewiththemoldandprogressestoattaincompletesolidificationinthesecondarycoolingzone,andthenthecaststeeliscuttoaprescribedlengthbyacutter.AcontinuouscasterisschematicallyillustratedinFig.2.2.2MoldlevelcontrolAsseeninFig.2,themoldsteelleveliscontinuouslymonitoredwithasensoranditsdeviationfromaprescribedtargetlevelisfedbacktoacontroller,whichoutputsasignaltoanactuatingcylindertoadjusttheopeningofaslidingnozzle.Theamountofmoltensteelflowintothemoldisthuscontrolledandthemoldsteelleveliscon-trolledtothetargetlevel.Thefluctuationofthemoldsteelleveliscausedbydisturbancesoriginatingfromthefillingandwithdrawalsystemsofacaster.Thedisturbanceoriginatingfromthefillingsystemmeans,morespecifi-cally,thechangeofmoltensteelflowcharacteristicsresultingfromcloggingofanentrynozzlewithnon-metallicinclusions,andthatoriginatingfromthewithdrawalsystemmeansperiodicalmoldsteellevelfluctuationresultingfrombulging(thermaldeformationofcaststeel)occurringinthewatercoolingzoneofacaster.3.MoldLevelControlModelThedevelopedmoldlevelcontrolmodelisexplainedhere.TheblockdiagramofthemodelwasshownearlierinFig.1,andtheexplanationshereafterarefocusedonthecomponentequationsofthephysicalmodelstoexpressthedynamicsoftheprocessinques-tion.Thedevelopedmodelbasedonmoltenflowanalysisisalsocomparedwithaconventionalintegralsystemmodel,andthediffer-encebetweenthetwoisclarified.3.1IntegralsystemmodelAnintegralsystemmodeltakesintoconsiderationonlythestaticvolumetricbalancebetweentheinfluxandoutfluxofmoltensteel,andassumesthatthedynamiccharacteristicofthemoldsteellevelcanbeexpressedintermsofasimpleintegralsystem.Inthiscase,thegoverningequationofthemoldsteellevelisgivenintheformofthefollowinglinearordinarydifferentialequation:(1)wherehistheheightofthemoldsteellevel(m),Aisthesectionalareaofamold(m2),Vinisthevolumetricinfluxrateofmoltensteel(m3/s),Voutisthevolumetricoutfluxrateofmoltensteel(m3/s),andtistime(s).Equation(1)meansthatthemoldsteellevelchangesinproportiontothevolumeofmoltensteel,anditgivesagoodap-proximationasfarastheturbulenceofmoltensteelflowlittleshowsonthesurface.However,inhigh-speedcasting,inwhichtheturbu-lenceofmoltensteelflowshowsitselfassurfaceripples,thiskindofmodelisnoteffectiveanylonger.Insuchacase,amodelbasedonmoltensteelflowanalysisasexplainedbelowisrequired.3.2MoltensteelflowanalysismodelAmoltensteelflowanalysismodelisbasedonatechniqueoffluiddynamicsandiscapableofaccuratelyexpressingthedynamiccharacteristicsofmoldsteellevelinconsiderationofturbulenceofmoltensteelflow.Whenthereisturbulenceofmoltensteelflow,themoltensteelflowinamoldisregardedasa3-dimensionalturbulentflowofanon-compressiblefluidhavingafreesurface,anditsgov-erningequationsaregivenintheformofthefollowingnon-linearpartialdifferentialequations:(2)(3)Equation(2)isthatofcontinuityandequation(3)thatofconser-vationofmomentum,whereuistheflowrateofmoltensteel(m/s),isthedensityofmoltensteel(kg/m3),Pispressure(N/m2),eiseffectivekinematicviscosity(m2/s),gisgravitationalacceleration(m/s2),andFisatermofanexternalforce(m/s2).Alargeeddysimu-lation(LES)modelisusedasaturbulencemodelinordertoexpressthedisturbance,orthetimedifference,ofaturbulentflow.Theposi-tionoftheboundaryofafreesurfaceisdefinedbythevolumeoffluid(VOF)method.Thetimedifferencesofmoltensteelflowrateandthefreesur-facearecalculatedbydiscretizingtheseequationsusingthecalculusoffinitedifferenceandnumericallysolvingthemusingtheiterativeanalysismethod.Fig.3showstheboundaryconditionsofamoltensteelflowanalysismodel.Aphysicalmodeltoaccuratelyexpressthemoltensteelflowinamoldandthedynamiccharacteristicsofawholemoltensteelsur-faceisthusconstructed.Thismodeliscapableofaccuratelyexpress-ingthedynamiccharacteristicsofmoltensteelsurfaceathigh-speedcasting.CylinderMoldControllerSlidingvalveTundishWatersprayRollsNozzleWithdrawaldirectionSensorFig.2Moldlevelcontrolincontinuouscastingprocessh=1AVinVoutdt0=uut+uu=1P+e2u+g+FNIPPONSTEELTECHNICALREPORTNo.89January2004-48-4.MoldLevelControlSimulationUsefulnessofthemoldlevelcontrolmodelbasedonmoltensteelflowanalysisinactualapplicationisexplainedbelowbasedonsimu-lationresults.Theresultsobtainedusinganintegralsystemmodelarealsoexplainedforcomparisonpurposes.TheconditionsfortheanalysisareshowninTable1.Thesimulationresultswereanalyzedunderaconditionofhigh-speedcastingforthepurposeofpredictingtheinfluenceofturbulenceofmoltensteelflowoverthemoldlevelcontrol.Assumingtheoccurrenceofbulgingasanexternalforce,amoldsteellevelfluctuationhavingacycletimeof10s(=afre-quencyof0.10Hz)wasimposed,themoltensteellevelwasmoni-toredwithasensor,andtheamountofmoltensteelinfluxwascon-trolledapplyingPIcontrol.4.1IntegralsystemmodelFig.4showsthetimefluctuationofdetectedmoltensteellevelcalculatedbythesimulationusinganintegralsystemmodel,andFig.5itspowerspectrum.Itisclearfromthefiguresthattheintegralsystemmodeldetectedonlyamoltensteellevelfluctuationhavingafrequencyof0.10Hzcausedbytheimposedexternalforce.4.2MoltensteelflowanalysismodelFig.6showsthetimefluctuationofdetectedmoltensteellevelcalculatedbythesimulationusingthemoltensteelflowanalysismodel,andFig.7itspowerspectrum.Itisclearfromthefiguresthatthemoltensteelflowanalysismodeldetectednotonlythelevelfluc-tuationof0.10Hzbutalsoahigh-frequencylevelfluctuationhavingafrequencyofapproximately0.70Hz.The0.70-Hzlevelfluctua-tionispresumedtorepresentastationarywaveinamold.Here,thefrequencyofthestationarywaveiscalculatedusingthefollowingtheoreticalequation:(4)wherefisfrequency(Hz),Nisaninteger(-),gisgravitationalacceleration(m/s2),andListhewidthofamold(m).WhenN=1andL=1,500,thenf=0.71Hzisobtainedfromequation(4),whichvalueagreeswellwiththefrequencyoftheripplingobtainedbythemoltensteelflowanalysismodel.ThefactthatsuchhighfrequencyripplingisobservedinactualCastingspeedSEN(rigidwall#1)Freesurface(VOF)Solidifiedshell(rigidwall#2)Casterbottom(outflowboundary)Nozzleinlet(inflowboundary)PouringvelocityFig.3BoundaryconditionsofmoltensteelflowanalysismodelDetectedlevel(mm)-20160150140130120110-15-100105-5Time(s)Power(normalized)(-)010.10.00.80.4Frequency(Hz)Moldwidth1,500mmMoldthikness240mmCastingspeed2.1mpmReferencelevel-5mmControlmethodPITable1AnalysisconditionsofmoldlevelcontrolsimulationDetectedlevel(mm)-20160150140130120110-15-100105-5Time(s)Power(normalized)(-)010.10.00.80.4Frequency(Hz)Fig.4DetectedmoltensteellevelcalculatedbyintegralsystemmodelFig.5FrequencycharacteristicofdetectedmoltensteellevelfluctuationcalculatedbyintegralsystemmodelFig.6DetectedmoltensteellevelcalculatedbymoltensteelflowanalysismodelFig.7Frequencycharacteristicofdetectedmoltensteellevelfluctuationcalculatedbymoltensteelflowanalysismodelf=12NgLcasteroperationservesasevidenceofthecapabilityofthemoltensteelflowanalysismodeltoaccuratelyreproducethedynamicchar-acteristicsofmoltensteellevel.Fig.8isanexamplemoldsteellevelchartofacommerciallyoperatedcaster,whereinripplinghavingafrequencyofroughly1.00Hzisdetected.Withthemoltensteelflowanalysismodel,itispossibletoevalu-atenotonlyadetectedmoldlevelbutalsoawholemoltensteelsur-faceinamold.Figs.9and10showthedistributionsoftheaverageandstandarddeviation,respectively,ofmoldsteellevelinmm,NIPPONSTEELTECHNICALREPORTNo.89January2004-49-Fig.12Distributionofstandarddeviationofmoltensteelflowrateatmoldthicknesscenter(analysis)-101086420-2-4-6-8310987654whereinthedarkerthearea,thelargerthevalue.ThewhiteareainFig.10istheareawherethemoldsteellevelisdetectedandcon-trolledwithasensor,anditisseeninthefigurethatthemoldsteellevelfluctuationiswellcontrollednearthewhitearea.Incontrast,themoldsteellevelfluctuationinotherareasisnotcontrolledandespeciallyintheoppositesidewherenosensorisprovided,thefluc-tuationislarge.Theaboveindicatesthatmoldlevelcontrolsup-presseslocallevelfluctuationsonly.Itisalsoseeninthefiguresthatthemoltensteellevelishighnearthemoldnarrowfacesandaroundthesubmergedentrynozzle,andthelevelfluctuationisalsolargeintheseareas.Thisisconsideredtoreflectthemoltensteelflowinthemold.Fig.11showsthedistribu-tionofthevectoroftimeaverageflowrateofmoltensteelatthethicknesscenterofamold.Themoltensteelflowingfromtheentrynozzlehitsthenarrowfacesandstrongupwardcirculatingflowsareformedthere.Here,itisseenthatthestrongupwardflowsalongthenarrowfacewallsliftthemoltensteelsurfacenearthem.Themoltensteellevelishighalsoaroundtheentrynozzle,becausethecirculat-ingflowsfromboththesidesrunintoeachotherthere,andasaresult,thelevelfluctuationisalsolargeinthisarea.Fig.12showsthedistributionofstandarddeviationofmoltensteelflowrateatthethicknesscenterofthesamemoldinm/s；thedarke