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1、SummaryManyconventionalrectangularbakingpanshaveaproblemthattheybakethecornersofthebatterfasterthantherestofthepan.Circularbakingpanseliminatethisproblem,buttakeupmorespaceintheoven.Weproposeasolutionthatgivenweightsforbakingconsistencyandspaceefficiencybasedonth
2、eimportanceofeachwillprovidetheoptimalbakingpanshape.WehavecomeupwithanalgorithmforsortingthepansofareaAeffectivelyandformedamodelthatdescribestheheatflowintothepanaswellasthebakingmixitself.Themodelwedevelopedwasbasedoffreal-lifedatathatwegatheredbybakingbrownie
3、susingcarefulmeasurement.Wedeterminedseveralphysicalproperties,suchasspecificheat,density,andheattransfercoefficientsforbrowniesdirectlythroughexperimentation.Wecomparedourresultswithpreviousstudiesonthepropertiesofcake-likefoodsandfoundthattheyweresimilar.Thenwe
4、ranthemodelusingthesecoefficientsandavarietyofdifferentplausiblebakingparameters,includingtemperatureandbakingtime.Wealsodevelopedatechniqueforoptimallyarrangingpolygonalshapedpansintheovenforanylengthto-widthratiotomaximizethenumberofpansthatcanfit.Forpolygons,w
5、econsideredbothlatticearrangementsandwhereapplicable,tessellatingpatterns.Thealgorithmforpackingalsotakesintoaccounttheneedforairflowbetweenthepansforoverallevencooking,whichmeansthatthepansarenotpackeddirectlynexttoeachothercausingunevenheatingonthepanslocatedne
6、arthesidesoftheoven.Theoverallefficiencyofapanshapewascalculatedusingthefollowingequation(wherepisaweightingfactor):Efftotal=p·Effspace+(1?p)·EffheatingUsingthealgorithmwefoundthatthetypeofpolygonneededtoefficientlyusetheovenspacewhileheatingthepanperimeterwasgre
7、atlydependentonthedimensionsoftheovenforweightingvaluesbetween0.2-1.0.(Below0.2,thecircularwww.madio.net數(shù)學(xué)中國(guó)panisveryheavilyfavoredregardlessoftheovendimensions.)Wealsofoundthataddingorsubtractingalittlebitfromthewidthorlengthoftheovencanhaveadrasticeffectonthesp
8、acingefficiencyratio.Whenwehavemaximallyhighefficiency,wecallthewidthandlengthofthepan”magicnumbers.”Forexample,iftheovendimensions(widthbylength)are30”x40”,wi
