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NIPPONSTEELTECHNICALREPORTNo.89January200446UDC621.746.27681.31EnvironmentProcessTechnologyCenter,TechnicalDevelopmentBureauFormulationofMoldLevelControlModelbyMoltenSteelFlowAnalysisMethodDaiSUZUKI1AbstractThispaperdescribesactivitiesrelatingtomoldlevelcontrolincontinuouscastingprocessascaseexamplesofourprocesscontrolsolutionbusiness,inwhichweclarifyandevaluatetheprocessphenomenaanddesignthecontrolsystem.Initially,amoldlevelcontrolmodelhasbeenformulatedbyfluidflowanalysistechniquethatevaluatesmoltensteelflowinthemoldquantitatively.Themodelcalculatesthefluidflowandfreesurfaceinthemoldbymoltensteelflowanalysismethod,whilecontrolsystemanalysisisperformedsimultaneously.Itenablesrealisticsimulationofthemoldlevelcontrolsystemconsideringthefluidflowturbulence.Asaconsequence,moldlevelcontrolinhighspeedcastingcanbepredictedandevaluatedwithhighaccuracy.1.IntroductionMoldlevelcontrolisafunctionoftheprocesscontrolofcontinuouscastingtomaintainthelevelofmoltensteelsurfaceinamoldhereinafterreferredtoasthemoldsteellevelconstant.Thishasasignificantinfluenceoverthequalityandyieldofthefinalproduct.Thereisaclosecorrelationbetweenthefluctuationofthemoldsteellevelandtheoccurrenceofsurfacedefectsoffinalproductsitisconsideredthat,whenthemoldsteellevelfluctuatessignificantly,castingpowderandotherimpuritiesfloatingonthesurfaceofmoltensteelareentrappedinsteelandtheyappearintheformofsurfacedefectsofsteelsheetproductsduringrolling.Topreventthis,moldlevelcontrolisdesignedsoastominimizethefluctuationofthemoldsteellevel.Thedynamicsofthemoldsteellevelis,roughlyspeaking,asimpleintegralsystemandforthisreason,theeffectofmoldlevelcontrolhasconventionallybeencalculatedinitsdevelopmentstageonanassumptionofanintegralsystem.Inpractice,however,disturbancesinmoltensteelflowtendtoshowintheformofsurfaceripplesasthecastingspeedincreases.Asaresult,withaconventionalcrudemodel,itwasdifficulttoobtainareliablesimulationresultanditwasimpossibletoaccuratelypredictandevaluatethecontrolperformanceofamoldlevelcontrolsystemunderaconditionofahighcastingspeed.Inviewofthesituation,theauthorhasformulatedamoldlevelcontrolmodelapplyingmoltensteelflowanalysis.Fig.1showsablockdiagramofthedevelopedmodel.Thismodelischaracterizedbydescribingthemoltensteelflowinamoldandthedynamicsofamoltensteelsurfacefromtheviewpointoffluidflowanalysisandcombiningthemwithcontrolsystemanalysis.Asaresult,amoldPlantmodelMoltensteelflowanalysisLevelcontrolFreesurfacedistributedSensorControllerDetectedlevelPouringvelocity1.3Dnumericalfluiddynamicsmethod2.ContinuityEq.,incompressibility3.NavierStokesEq.4.Finitedifferencemethod5.TurbulencemodelLargeeddysimulationmodel6.FreesurfaetrackingVolumeoffluidmethodFig.1MoldlevelcontrolmodelbasedonmoltensteelflowanalysisNIPPONSTEELTECHNICALREPORTNo.89January200447levelcontrolsimulationinconsiderationoftheturbulenceofmoltensteelflowwasmadeviable,anditbecamepossibletoaccuratelypredictandevaluatetheperformanceofmoldlevelcontrolatahighcastingspeed.2.ContinuousCastingProcessBeforediscussingthemoldlevelcontrolmodel,continuouscastingequipmentandmoldlevelcontrol,whichconstitutethebackgroundofthemodel,arebrieflyexplained.2.1ContinuouscastingequipmentContinuouscastingequipmentaresteelproductionfacilitiesforefficientlyproducingslabsorbloomsorbilletsforthesubsequentrollingprocessbycontinuouslysolidifyingmoltensteelafteritsrefininginasteelmakingfurnacesuchasaconverter.Moltensteeldischargedfromaladleistemporarilystoredinatundishandthenpouredthroughanimmersionentrynozzleintoamoldtheinsidewallsofwhicharelinedwithwatercooledcopperplates.Thesolidificationofthecaststeelbeginsatitsinterfacewiththemoldandprogressestoattaincompletesolidificationinthesecondarycoolingzone,andthenthecaststeeliscuttoaprescribedlengthbyacutter.AcontinuouscasterisschematicallyillustratedinFig.2.2.2MoldlevelcontrolAsseeninFig.2,themoldsteelleveliscontinuouslymonitoredwithasensoranditsdeviationfromaprescribedtargetlevelisfedbacktoacontroller,whichoutputsasignaltoanactuatingcylindertoadjusttheopeningofaslidingnozzle.Theamountofmoltensteelflowintothemoldisthuscontrolledandthemoldsteelleveliscontrolledtothetargetlevel.Thefluctuationofthemoldsteelleveliscausedbydisturbancesoriginatingfromthefillingandwithdrawalsystemsofacaster.Thedisturbanceoriginatingfromthefillingsystemmeans,morespecifically,thechangeofmoltensteelflowcharacteristicsresultingfromcloggingofanentrynozzlewithnonmetallicinclusions,andthatoriginatingfromthewithdrawalsystemmeansperiodicalmoldsteellevelfluctuationresultingfrombulgingthermaldeformationofcaststeeloccurringinthewatercoolingzoneofacaster.3.MoldLevelControlModelThedevelopedmoldlevelcontrolmodelisexplainedhere.TheblockdiagramofthemodelwasshownearlierinFig.1,andtheexplanationshereafterarefocusedonthecomponentequationsofthephysicalmodelstoexpressthedynamicsoftheprocessinquestion.Thedevelopedmodelbasedonmoltenflowanalysisisalsocomparedwithaconventionalintegralsystemmodel,andthedifferencebetweenthetwoisclarified.3.1IntegralsystemmodelAnintegralsystemmodeltakesintoconsiderationonlythestaticvolumetricbalancebetweentheinfluxandoutfluxofmoltensteel,andassumesthatthedynamiccharacteristicofthemoldsteellevelcanbeexpressedintermsofasimpleintegralsystem.Inthiscase,thegoverningequationofthemoldsteellevelisgivenintheformofthefollowinglinearordinarydifferentialequation1wherehistheheightofthemoldsteellevelm,Aisthesectionalareaofamoldm2,Vinisthevolumetricinfluxrateofmoltensteelm3/s,Voutisthevolumetricoutfluxrateofmoltensteelm3/s,andtistimes.Equation1meansthatthemoldsteellevelchangesinproportiontothevolumeofmoltensteel,anditgivesagoodapproximationasfarastheturbulenceofmoltensteelflowlittleshowsonthesurface.However,inhighspeedcasting,inwhichtheturbulenceofmoltensteelflowshowsitselfassurfaceripples,thiskindofmodelisnoteffectiveanylonger.Insuchacase,amodelbasedonmoltensteelflowanalysisasexplainedbelowisrequired.3.2MoltensteelflowanalysismodelAmoltensteelflowanalysismodelisbasedonatechniqueoffluiddynamicsandiscapableofaccuratelyexpressingthedynamiccharacteristicsofmoldsteellevelinconsiderationofturbulenceofmoltensteelflow.Whenthereisturbulenceofmoltensteelflow,themoltensteelflowinamoldisregardedasa3dimensionalturbulentflowofanoncompressiblefluidhavingafreesurface,anditsgoverningequationsaregivenintheformofthefollowingnonlinearpartialdifferentialequations23Equation2isthatofcontinuityandequation3thatofconservationofmomentum,whereuistheflowrateofmoltensteelm/s,ρisthedensityofmoltensteelkg/m3,PispressureN/m2,νeiseffectivekinematicviscositym2/s,gisgravitationalaccelerationm/s2,andFisatermofanexternalforcem/s2.AlargeeddysimulationLESmodelisusedasaturbulencemodelinordertoexpressthedisturbance,orthetimedifference,ofaturbulentflow.ThepositionoftheboundaryofafreesurfaceisdefinedbythevolumeoffluidVOFmethod.Thetimedifferencesofmoltensteelflowrateandthefreesurfacearecalculatedbydiscretizingtheseequationsusingthecalculusoffinitedifferenceandnumericallysolvingthemusingtheiterativeanalysismethod.Fig.3showstheboundaryconditionsofamoltensteelflowanalysismodel.Aphysicalmodeltoaccuratelyexpressthemoltensteelflowinamoldandthedynamiccharacteristicsofawholemoltensteelsurfaceisthusconstructed.Thismodeliscapableofaccuratelyexpressingthedynamiccharacteristicsofmoltensteelsurfaceathighspeedcasting.CylinderMoldControllerSlidingvalveTundishWatersprayRollsNozzleWithdrawaldirectionSensorFig.2Moldlevelcontrolincontinuouscastingprocessh1AVin–Voutdt0∇⋅u∂u∂tu⋅∇u–1ρ∇Pνe∇2ugFNIPPONSTEELTECHNICALREPORTNo.89January2004484.MoldLevelControlSimulationUsefulnessofthemoldlevelcontrolmodelbasedonmoltensteelflowanalysisinactualapplicationisexplainedbelowbasedonsimulationresults.Theresultsobtainedusinganintegralsystemmodelarealsoexplainedforcomparisonpurposes.TheconditionsfortheanalysisareshowninTable1.Thesimulationresultswereanalyzedunderaconditionofhighspeedcastingforthepurposeofpredictingtheinfluenceofturbulenceofmoltensteelflowoverthemoldlevelcontrol.Assumingtheoccurrenceofbulgingasanexternalforce,amoldsteellevelfluctuationhavingacycletimeof10safrequencyof0.10Hzwasimposed,themoltensteellevelwasmonitoredwithasensor,andtheamountofmoltensteelinfluxwascontrolledapplyingPIcontrol.4.1IntegralsystemmodelFig.4showsthetimefluctuationofdetectedmoltensteellevelcalculatedbythesimulationusinganintegralsystemmodel,andFig.5itspowerspectrum.Itisclearfromthefiguresthattheintegralsystemmodeldetectedonlyamoltensteellevelfluctuationhavingafrequencyof0.10Hzcausedbytheimposedexternalforce.4.2MoltensteelflowanalysismodelFig.6showsthetimefluctuationofdetectedmoltensteellevelcalculatedbythesimulationusingthemoltensteelflowanalysismodel,andFig.7itspowerspectrum.Itisclearfromthefiguresthatthemoltensteelflowanalysismodeldetectednotonlythelevelfluctuationof0.10Hzbutalsoahighfrequencylevelfluctuationhavingafrequencyofapproximately0.70Hz.The0.70Hzlevelfluctuationispresumedtorepresentastationarywaveinamold.Here,thefrequencyofthestationarywaveiscalculatedusingthefollowingtheoreticalequation4wherefisfrequencyHz,Nisaninteger,gisgravitationalaccelerationm/s2,andListhewidthofamoldm.WhenN1andL1,500,thenf0.71Hzisobtainedfromequation4,whichvalueagreeswellwiththefrequencyoftheripplingobtainedbythemoltensteelflowanalysismodel.ThefactthatsuchhighfrequencyripplingisobservedinactualCastingspeedSENrigidwall1FreesurfaceVOFSolidifiedshellrigidwall2CasterbottomoutflowboundaryNozzleinletinflowboundaryPouringvelocityFig.3BoundaryconditionsofmoltensteelflowanalysismodelDetectedlevelmm20160150140130120110151001055TimesPowernormalized010.10.010.20.61.00.80.4FrequencyHzMoldwidth1,500mmMoldthikness240mmCastingspeed2.1mpmReferencelevel5mmControlmethodPITable1AnalysisconditionsofmoldlevelcontrolsimulationDetectedlevelmm20160150140130120110151001055TimesPowernormalized010.10.010.20.61.00.80.4FrequencyHzFig.4DetectedmoltensteellevelcalculatedbyintegralsystemmodelFig.5FrequencycharacteristicofdetectedmoltensteellevelfluctuationcalculatedbyintegralsystemmodelFig.6DetectedmoltensteellevelcalculatedbymoltensteelflowanalysismodelFig.7Frequencycharacteristicofdetectedmoltensteellevelfluctuationcalculatedbymoltensteelflowanalysismodelf12πNπgLcasteroperationservesasevidenceofthecapabilityofthemoltensteelflowanalysismodeltoaccuratelyreproducethedynamiccharacteristicsofmoltensteellevel.Fig.8isanexamplemoldsteellevelchartofacommerciallyoperatedcaster,whereinripplinghavingafrequencyofroughly1.00Hzisdetected.Withthemoltensteelflowanalysismodel,itispossibletoevaluatenotonlyadetectedmoldlevelbutalsoawholemoltensteelsurfaceinamold.Figs.9and10showthedistributionsoftheaverageandstandarddeviation,respectively,ofmoldsteellevelinmm,NIPPONSTEELTECHNICALREPORTNo.89January200449Fig.12Distributionofstandarddeviationofmoltensteelflowrateatmoldthicknesscenteranalysis1010864202468310987654whereinthedarkerthearea,thelargerthevalue.ThewhiteareainFig.10istheareawherethemoldsteellevelisdetectedandcontrolledwithasensor,anditisseeninthefigurethatthemoldsteellevelfluctuationiswellcontrollednearthewhitearea.Incontrast,themoldsteellevelfluctuationinotherareasisnotcontrolledandespeciallyintheoppositesidewherenosensorisprovided,thefluctuationislarge.Theaboveindicatesthatmoldlevelcontrolsuppresseslocallevelfluctuationsonly.Itisalsoseeninthefiguresthatthemoltensteellevelishighnearthemoldnarrowfacesandaroundthesubmergedentrynozzle,andthelevelfluctuationisalsolargeintheseareas.Thisisconsideredtoreflectthemoltensteelflowinthemold.Fig.11showsthedistributionofthevectoroftimeaverageflowrateofmoltensteelatthethicknesscenterofamold.Themoltensteelflowingfromtheentrynozzlehitsthenarrowfacesandstrongupwardcirculatingflowsareformedthere.Here,itisseenthatthestrongupwardflowsalongthenarrowfacewallsliftthemoltensteelsurfacenearthem.Themoltensteellevelishighalsoaroundtheentrynozzle,becausethecirculatingflowsfromboththesidesrunintoeachotherthere,andasaresult,thelevelfluctuationisalsolargeinthisarea.Fig.12showsthedistributionofstandarddeviationofmoltensteelflowrateatthethicknesscenterofthesamemoldinm/sthedarkerthearea,thelargerthevalue.Itisseeninthefigurethatthewholemoltensteelinthemoldissignificantlydisturbedbytheinfluxfromthenozzle.Itisconsideredthattheturbulencebecomessignificantasthespeedoftheincomingflowbecomeshigh,showingitselfastheripplingofthemoltensteelsurface.Asstatedabove,ithasbeendemonstratedthatamoltensteelflowanalysismodelenablesarealisticmoldlevelcontrolsimulationreflectingtheturbulenceofmoltensteelflow.5.ClosingAmoldlevelcontrolmodelapplyingmoltensteelflowanalysishasbeenformulated,andthusarealisticmoldlevelcontrolsimulationreflectingtheturbulenceofmoltensteelflowhasbeenmadepossible.Theeffectivenessofthedevelopedmodelhasbeenconfirmedthroughsimulations.Amoldlevelcontrolsystembasedonthedevelopedmodelwillbedesignedforapplicationtohighspeedcastingoperation,anditseffectivenesswillbeevaluatedthroughtestsinactualoperation.References1Suzuki,D.etal.LevelControlModelbyNumericalFluidDynamicsMethod.ProceedingsoftheFourthInternationalConferenceonIntelligentProcessingandManufacturingofMaterials.IPMM03.2003CDROMFig.8MoldsteellevelchartincommercialoperationFig.9AveragemoltensteellevelinmoldanalysisFig.10Standarddeviationofmoltensteellevelinmoldanalysis2.52.01.51.000.51.01.5NXFig.11Distributionoftimeaverageofmoltensteelflowrateatmoldthicknesscenteranalysis2.52.01.51.000.51.01.5NX00.40.30.250.20.150.10.050.35
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