2016年美赛A题O奖论文

2016MathematicalContestinModeling(MCM/ICM)SummarySheetSummaryAtraditionalbathtubcannotbereheatedbyitself,sousershavetoaddhotwaterfromtimetotime.Ourgoalistoestablishamodelofthetemperatureofbathwaterinspaceandtime.Thenweareexpectedtoproposeanoptimalstrategyforuserstokeepthetemperatureevenandclosetoinitialtemperatureanddecreasewaterconsumption.Tosimplifymodelingprocess,wefirstlyassumethereisnopersoninthebathtub.Weregardthewholebathtubasathermodynamicsystemandintroduceheattransferformulas.Weestablishtwosub-models:addingwaterconstantlyanddiscontinuously.Asfortheformersub-model,wedefinethemeantemperatureofbathwater.IntroducingNewtoncoolingformula,wedeterminetheheattransfercapacity.Afterderivingthevalueofparameters,wededuceformulastoderiveresultsandsimulatethechangeoftemperaturefieldviaCFD.Asforthesecondsub-model,wedefineaniterationconsistingoftwoprocess:heatingandstandby.Accordingtoenergyconservationlaw,weobtaintherelationshipoftimeandtotalheatdissipatingcapacity.Thenwedeterminethemassflowandthetimeofaddinghotwater.WealsouseCFDtosimulatethetemperaturefieldinsecondsub-model.Inconsiderationofevaporation,wecorrecttheresultsofsub-modelsreferringtosomescientists’studies.Wedefinetwoevaluationcriteriaandcomparethetwosub-models.Addingwaterconstantlyisfoundtokeepthetemperatureofbathwaterevenandavoidwastingtoomuchwater,soitisrecommendedbyus.Thenwedeterminetheinfluenceofsomefactors:radiationheattransfer,theshapeandvolumeofthetub,theshape/volume/temperature/motionsoftheperson,thebubblesmadefrombubblebathadditives.Wefocusontheinfluenceofthosefactorstoheattransferandthenconductsensitivityanalysis.Theresultsindicatesmallerbathtubwithlesssurfacearea,lighterpersonalmass,lessmotionsandmorebubbleswilldecreaseheattransferandsavewater.Basedonourmodelanalysisandconclusions,weproposetheoptimalstrategyfortheuserinabathtubandexplainthereasonofuneventemperaturethroughoutthebathtub.Inaddition,wemakeimprovementforapplyingourmodelinreallife.Keywords:HeattransferThermodynamicsystemCFDEnergyconservationForofficeuseonlyT1T2T3T4ForofficeuseonlyF1F2F3F4TeamControlNumber44398ProblemChosenATeam#44398Page2of51EnjoyaCozyandGreenBathContents1Introduction..................................................................................................41.1Background.........................................................................................41.2LiteratureReview...............................................................................41.3RestatementoftheProblem..............................................................42AssumptionsandJustification..................................................................63Notations........................................................................................................74ModelOverview...........................................................................................75Sub-modelI:AddingWaterContinuously............................................85.1ModelEstablishment.........................................................................95.1.1ControlEquationsandBoundaryConditions....................95.1.2DefinitionoftheMeanTemperature..................................115.1.3DeterminationofHeatTransferCapacity.........................115.2Results................................................................................................135.2.1DeterminationofParameters..............................................135.2.2CalculatingResults...............................................................145.2.3SimulatingResults................................................................156Sub-modelII:AddingWaterDiscontinuously....................................186.1HeatingModel..................................................................................186.1.1ControlEquationsandBoundaryConditions..................186.1.2DeterminationofInflowTimeandAmount.....................196.2StandbyModel.................................................................................206.2.1ProcessAnalysis....................................................................206.2.2CalculationofParameters....................................................206.3Results................................................................................................216.3.1DeterminationofParameters..............................................216.3.2CalculatingResults...............................................................23Team#44398Page3of516.3.3SimulatingResults................................................................236.4Conclusion........................................................................................277CorrectionandContrastofSub-Models...............................................277.1CorrectionwithEvaporationHeatTransfer................................277.1.1CorrectionPrinciple..............................................................277.1.2CorrectionResults.................................................................287.2ContrastofTwoSub-M