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1、TeamControlNumberForofficeuseonlyForofficeuseonly44398T1F1T2ProblemChosenF2T3F3T4AF42016MathematicalContestinModeling(MCM/ICM)SummarySheetSummaryAtraditionalbathtubcannotbereheatedbyitself,sousershavetoaddhotwaterfromtimetotime.Ourgoalistoestablishamodelofthetemperatureofbat
2、hwaterinspaceandtime.Thenweareexpectedtoproposeanoptimalstrategyforuserstokeepthetemperatureevenandclosetoinitialtemperatureanddecreasewaterconsumption.Tosimplifymodelingprocess,wefirstlyassumethereisnopersoninthebathtub.Weregardthewholebathtubasathermodynamicsystemandintrod
3、uceheattransferformulas.Weestablishtwosub-models:addingwaterconstantlyanddiscontinuously.Asfortheformersub-model,wedefinethemeantemperatureofbathwater.IntroducingNewtoncoolingformula,wedeterminetheheattransfercapacity.Afterderivingthevalueofparameters,wededuceformulastoderiv
4、eresultsandsimulatethechangeoftemperaturefieldviaCFD.Asforthesecondsub-model,wedefineaniterationconsistingoftwoprocess:heatingandstandby.Accordingtoenergyconservationlaw,weobtaintherelationshipoftimeandtotalheatdissipatingcapacity.Thenwedeterminethemassflowandthetimeofadding
5、hotwater.WealsouseCFDtosimulatethetemperaturefieldinsecondsub-model.Inconsiderationofevaporation,wecorrecttheresultsofsub-modelsreferringtosomescientists’studies.Wedefinetwoevaluationcriteriaandcomparethetwosub-models.Addingwaterconstantlyisfoundtokeepthetemperatureofbathwat
6、erevenandavoidwastingtoomuchwater,soitisrecommendedbyus.Thenwedeterminetheinfluenceofsomefactors:radiationheattransfer,theshapeandvolumeofthetub,theshape/volume/temperature/motionsoftheperson,thebubblesmadefrombubblebathadditives.Wefocusontheinfluenceofthosefactorstoheattran
7、sferandthenconductsensitivityanalysis.Theresultsindicatesmallerbathtubwithlesssurfacearea,lighterpersonalmass,lessmotionsandmorebubbleswilldecreaseheattransferandsavewater.Basedonourmodelanalysisandconclusions,weproposetheoptimalstrategyfortheuserinabathtubandexplainthereaso
8、nofuneventemperaturethroughoutthebathtub.Inaddition,wemakeimprovementforapp
