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1、ForofficeuseonlyT1________________T2________________T3________________T4________________TeamControlNumber18486ProblemChosenAForofficeuseonlyF1________________F2________________F3________________F4________________AbstractThedifficultyofevaluatingtheperformanceofab
2、rowniepanmainlyliesinhowtoquantifyitstwocrucialaspects--theevennessofheatdistributionandthespatialusageofoven,andlinkthemtogethertogiveacomprehensiverecommendationofshape.Weanalyzeeachoftheaspectsnumerically,andfinallysuggestthatroundedrectangularpansarethebestto
3、meetbothconditions.Firstly,westudytheheatdistributionbytwomodels,atemperaturemodelbasedonheattransfertheory,andastandarddeviationmodelevaluatingevenness.Tosetupthetemperaturemodel,weapproachthe3-dimensionalproblemfromthe1-dimensionalsituation,andusethetypicalmult
4、iplicationmethod,basedonFourier’sLaw.Weanalyzetheair,thepan,andthecake.Usingtheexpressionoftemperature,wedrawthefigureofitsdistributionineachdifferentpan.Itshowshowheatisdistributedattheedgeandinside.Thestandarddeviation(STD)modelcomparesthestandarddeviationofthe
5、accumulatingheatindifferentpans.Acompletemodelmusttakeintoaccounttheheataccumulationinaperiodbecauseovercookingdoesn’thappenataninstant,anditisaresultoflastingunevenheatdistribution.Afteramodification,ourmodel,whichisbasedonthewhole20-minutesprocessofbaking,ismor
6、evalidonevaluatingevennessofheatdistribution.Itsuggeststhataroundpandistributesheatmostevenly.Secondly,wecalculatethespatialusagebyanalyticgeometrymethod.Undoubtedly,arectangularpanusesspacemostefficiently,andaroundpanviceversa.However,arectangularpanistheworstin
7、termsofheatdistribution.Inordertofindapangoodenoughinbothaspectsmentioned,weconsiderothershapesinbetween.First,wedenythepossibilityofellipseasanidealchoiceaftersomecalculation.Thenweanalyzeroundedrectanglefrombothaspectswithstatistics.Tocombinebothconditionsrequi
8、red,wecreateanobjectivefunction,involvingweightsofboth.ThespatialusageratioandtheSTDaremainpartsofthefunction.Withtheweightandthewidth-lengthratiogiven,themaxi
