最新年美赛A题H奖58265390.docx
《最新年美赛A题H奖58265390.docx》由会员分享,可在线阅读,更多相关《最新年美赛A题H奖58265390.docx(24页珍藏版)》请在冰豆网上搜索。
最新年美赛A题H奖58265390
年美赛A题H奖58265390
Summary
Webuildtwobasicmodelsforthetwoproblemsrespectively:
oneistoshowthedistributionofheatacrosstheouteredgeofthepanfordifferentshapes,rectangular,circularandthetransitionshape;anotheristoselectthebestshapeforthepanundertheconditionoftheoptimizationofcombinationsofmaximalnumberofpansintheovenandthemaximalevenheatdistributionoftheheatforthepan.
Wefirstusefinite-differencemethodtoanalyzetheheatconductandradiationproblemandderivetheheatdistributionoftherectangularandthecircular.Intermsofourisothermalcurveoftherectangularpan,weanalyzetheheatdistributionofroundedrectanglethoroughly,usingfinite-elementmethod.Wethenusenonlinearintegerprogrammingmethodtosolvethemaximalnumberofpansintheoven.Intheevenheatdistribution,wedefineafunctiontoshowthedegreeoftheevenheatdistribution.WeusepolynomialfittingwithmultiplevariablestosolvetheobjectivefunctionForthelastproblem,combiningtheresultsabove,weanalyzehowresultsvarywiththedifferentvaluesofwidthtolengthratioW/Landtheweightfactorp.Atlast,wevalidatethatourmethodiscorrectandrobustbycomparingandanalyzingitssensitivityandstrengths/weaknesses.
Basedontheworkabove,weultimatelyputforwardthattheroundedrectangularshapeisperfectconsideringoptimalnumberofthepansandevenheatdistribution.AndanadvertisementispresentedfortheBrownieGourmetMagazine.
Contents
1Introduction3
1.1Browniepan3
1.2Background3
1.3ProblemDescription3
2.Modelforheatdistribution3
2.1Problemanalysis3
2.2Assumptions4
2.3Definitions4
2.4Themodel4
3Resultsofheatdistribution7
3.1Basicresults7
3.2Analysis9
3.3Analysisofthetransitionshape—roundedrectangular9
4Modeltoselectthebestshape11
4.1Assumptions11
4.2Definitions11
4.3Themodel12
5ComparisionandDegreeoffitting19
6Sensitivity20
7Strengths/weaknesses21
8Conclusions21
9AdvertisementfornewBrownieMagazine23
10References24
1Introduction
1.1Browniepan
TheBrowniePanisusedtomakeBrownieswhichareakindofpopularcakesinAmerica.Itusuallyhasmanylatticesinitandismadeofmetalorothermaterialstoconductheatwell.Itistrivially9×9inchor9×13inchinsize.OneexampleoftheconcreteshapeofBrowniepanisshowninFigure1
Figure1theshapeofBrowniePan(source:
GoogleImage)
1.2Background
BrowniesaredeliciousbuttheBrowniePanhasafetaldrawback.Whenbakinginarectangularpan,thefoodcaneasilygetovercookedinthe4corners,whichisveryannoyingforthegreedygourmets.Inaroundpan,theheatisevenlydistributedovertheentireouteredgebutisnotefficientwithrespecttousinginthespaceinanoven,whichmostcakebakerswouldnotliketosee.Soourgoalistoaddressthisproblem.
1.3ProblemDescription
Firstly,weareaskedtodevelopamodeltoshowthedistributionofheatacrosstheouteredgeofapanfordifferentshapes,fromrectangulartocircularincludingthetransitionshapes;thenwewillbuildanothermodeltoselectthebestshapeofthepanfollowingtheconditionoftheoptimizationofcombinationsofmaximalnumberofpansintheovenandmaximalevendistributionofheatforthepan.
2.Modelforheatdistribution
2.1Problemanalysis
Hereweuseafinitedifferencemodeltoillustratethedistributionofheat,andithasbeenextensivelyusedinmodelingforitscharacteristicabilitytohandleirregulargeometriesandboundaryconditions,spatialandtemporalpropertiesvariations.Inliterature1,sampleswitharectangulargeometricformaredifficulttoheatuniformly,particularlyatthecornersandedges.Theythinkmicrowaveradiationintheovencanbecrudelythoughtofasimpingingonthesamplefromall,whichwegenerallyacknowledge.Buttheyemphasizetherotation.
Generally,whenbakingintheoven,thecakesabsorbheatbythreeways:
thermalradiationofthepipesintheoven,heatconductionofthepan,andairconvectionintheoven.Consideringthattheinfluenceofconvectionissmall,weassumeitnegligible.Soweonlytakethermalradiationandconductionintoaccount.Theheatistransferredfromtheoutsidetotheinsidewhilewaterinthecakeisonthecontrary.Thetemperatureoutsideincreasemorerapidlythanthatinside.Andthecontactareabetweenthepanandtheoutsidecakeislargerthanthatbetweenthepanandtheinsidecakes,whichillustratewhycakesinthecornergetovercookedeasily.
2.2Assumptions
●Wetakethepanandcakesasblackbody,sotheabsorptionofheatineachareaunitandtimeunitisthesame,whichdrasticallysimplifiesourcalculation.
●Weassumetheairconvectionnegligible,consideringitscomplexityandthesmallinfluenceonthetemperatureincrease.
●Weneglecttheevaporationofwaterinsidethecake,whichmayimpedetheincreaseoftemperatureofcakes.
●Weignorethethicknessofcakesandthepan,sothemodelwebuildistwo-dimensional.
2.3Definitions
Φ:
heatflowsintothenode
Q:
theheattakeninbycakesorpansfromtheheatpipes
:
energyincreaseofeachcakeunit
:
energyincreaseofthepanunit
:
temperatureatmomentiandpoint(m,n)
C1:
thespecificheatcapacityofthecake
C2:
thespecificheatcapacityofthepan
:
temperatureofthepanatmomenti
T1:
temperatureintheoven,whichweassumeisaconstant
2.4Themodel
Hereweusefinite-differencemethodtoderivetherelationshipoftemperaturesattimei-1andtimeiatdifferentplaceandtherelationshipoftemperaturesbetweenthepanandthecake.
Firstwedivideacakeintosmallunits,whichcanbeexpressedbyametric.Inthefollowingsection,wewilldiscussthecakeunitindifferentplacesofthepan.
Step1;temperaturesofcakesinterior
Figure2heatflow
Accordingtoenergyconversationprinciple,wecanget
(2.4.1)
ConsideringFourierLawand△x=△y,weget
(2.4.2)
AccordingtoStefan-BoltzmanLaw,
(2.4.3)
WhereAistheareacontacting,cistheheatconductance.σistheStefan-Boltzmannconstant,andequals5.73×108Jm-2s-1k-4.
(2.4.4)
Substituting(2.4.2)-(2.4.4)into(2.4.1),weget
Thisequationdemonstratestherelationshipoftemperatureatmomentiandmomenti-1aswellastherelationshipoftemperatureat(m,n)anditssurroundingpoints.
Step2:
temperatureofthecakeouterandthepan
●Forthe4corners
Figure3therelativepositionofthecakeandthepaninthefirstcorner
Becausethecontactingareaistwotimes,weget
●Foreveryedge
Figure4therelativepositionofthecakeandthepanattheedge
Similarly,wederive
Nowthatwehavederivedtheexpressoftemperaturesofcakesbothtemporallyandspatially,wecanuseiterationtogetthecurveoftemperaturewiththevariables,timeandlocation.
3Resultsofheatdistribution
3.1Basicresults
●Rectangular
Preliminarily,wefocusononecorneronly.Afterrunningtheprogramme,weobtainthefollowingfigure.
Figure5heatdistributionatonecorner
Figure5demonstratesthetemperatureatthecornerishigherthanitssurroundingpoints,that’swhyfoodatcornersgetovercookedeasily.
Thenweiterateglobally,andgetFigure6.
Figure6heatdistributionintherectangularpan
Figure6canintuitivelyillustratesthetemperatureatcornersisthehighest,andtemperatureontheedgeislesshigherthanthatatcorners,butismuchhigherthanthatatinteriorpoints,whichsuccessfullyexplainstheproblem“productsgetovercookedatthecornersbuttoalesserextentontheedge”.
Afterdrawingtheheatdistributionintwodimensions,wesamplesomepointsfromtheinsidetotheoutsideinarectangularandobtaintherelationshipbetweentemperatureanditerationtimes,whichisshowninFigure7
Figure7
FromFigure7,thetemperaturesgoupwithtimegoingandthenkeepnearlyparalleltothex-axis.Ontheotherhand,temperatureatthecenterascendstheslowest,thenedgeandcorner,whichmeansgivencookingtime,foodatthecenterofthepaniscookedjustwellwhilefoodatthecornerofthepanhasalreadygetovercooked,butalesserextenttotheedge.
●Round
Weuseourmodeltoanalyzetheheatdistributioninaround,justadaptingtherectangularunitsintosmallannuluses,byrunningourprogramme,wegetthefollowingfigure.
Figure8theheatdistributioninthecirclepan
Figure8showsheatdistributionincircleareaiseven,theproductsattheedgearecookedtothesameextentapproximately.
3.2Analysis
Finally,wedrawtheisothermalcurveofthepan.
●Rectangular
Figure9theisothermalcurveoftherectangularpan
Figure9demenstratestheisothermallinesarealmostconcentriccirclesinthecenterofthepanandbecomeroundedrectanglesouter,whichprovidestheorysupportforfollowinganalysis..
●Circular
Figure10theisothermalcurveofcircular
Theisothermalcurvesofthecircularareseriesofconcentriccircles,demonstratingthattheheatisevendistributed.
3.3Analysisofthetransitionshape—roundedrectangular
Fromtheaboveanalysis,wefindthattheisothermalcurvearenearlyroundedrectangularsintherectangularpan,soweperspectivethetransitionshapebetweenrectangularandcircularisroundedrectangular,consideringtheefficiencyofusingspaceoftheovenandtheevenheatdistribution.Inthefollowingsection,wewillanalyzetheheatdistributioninroundedrectangularpanusingfiniteelementapproach.
Duringthecookingprocess,thetemperaturegoesupgradually.Butatacertainmoment,thetemperaturecanbeassumedaconstant.SotheboundaryconditionyieldsDirichletboundary