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ummryWe bil to asicmodes or t poemrpectiely: onesto show hedistribution of hat crs the outer de of t pan fr diert ses, ectaa,icul and e transition shape; othr is o slect hebs shpeothe pan under th conditi f the oimizaton of omiatos ofmaxia number of pasine ov an the xmal eve heatdistrbutn f theheat o the an.We irstuse fintediferece method t nalyzethehatondut nd rdiatipoblem and drive te h dtributi of theectangur and the iclarI ters of our isothermalcurv of the rctangular pn, e aalyzethe het dtrbtionf rundedrectale horughy, usn finieelmntmethd. Wethe u noninear iteg progrmig eho to sl the mamal umber ofpasin theoen。In the vheatditribio, w defin afcton how dgree of evn heat distributi. We polynomalfitting with tiple varibles o sovethe jetiefunction Fo e lat prole, combini te rels above,we any how esults vaywit he dffentales f idt to lngt ratioW/ adeweigt fat p。 Atast, we aidtthat our mthod i core and robut by comparing and analyzn is nstivi and srengths/weanese Basd o th work aove, e utimately pufowrd tatrned recngula shei prfect conide timal uber f th pas and even het distrbion.And n artsenispresente o t BwnieGumet Maazine.ontents1ntducion311Bowniepa3。2Bcud31.Poblem Deciption32.Model foheat istributio。1 roblm analyss32。 Asmptns43 Dfnitions2。 e model4esul f hetditribution73。 Basic rsl7。Analysis3.alyis of the rasitio shaerodd rectngular4 Modl to select te t shap111 Asumptin114.2 Dfitns114。3 Themde5 Comaisin and Der ofittg96 Snstivt2Strets/weakesss8 Clsios219 Advertisemetfor new BrwneMagazin231 Reerences41ntrductin1.1 rownieanThe rwne Pn isused o make Brownie whih are knd o popular ckesin Arca.It sallyhsma ttic iandis ma of metal or othe atial ondct heat wel。 istrvially99 inch or913nc n size。 n exmple of the orete spe f Brwnie panisson i igreFigure the shpe ofBowie Pn (soure: GoogleImge)1.2 acgrun Bres ae deliius bt te Bownie Pan has a fetal rwback Whenaking n a rectagulr , he food ca esy g overookednthe 4 corers, wichisvry annoyi r the greedy gourmetsIn a roun pn, te hea is evenlydistributed over th etie oe eg but is not efficienwith rpett usig in te pae n ove, wic mos cake rs wold nt ike oee So ur goa is to dresstsrobl.1.3 Probem DesritonFirsy, we ar askedto deelop ode to howthedtributionohea cro theoteredg ofa pan fordiffent shap,o rtangular o circularicluing thtrasiti ape; the we will buldanor moe select the bes hap of the panfolowing the codiinof he oimizationof combion of aialnumbr f pansin te ven an mximal even srbuto of at fr the pan.2 Mofr hat distrbution2。1 Prblem nalysisHre we use a finie dfferece odl to ilstae h disrbuon of at,d it ha been exensivelued in modeling or its haraeristic bilty t hnde irrular emetie and boundary oditios, spai and temporapropetievariatns Shixiong Liu, Mika Fukuoka, Noboru Sakai, A finite element model for simulating temperature distributions in rotating food during microwave heating, Journal of food engineering, Volume 115, issue 1, March 2013 Page 49-62。 In litratr 1, smpleswih a reangl emetricform are dificul tohetunifomly,particularl t hornersan edg。 hy think icowav raiaion n the on cn e crdelythught ofas impiing o tesample from all, whch e eeaaowed Butthy empasizetheroation.Geerall, whn akin in the oven, he cae abo heat y thre ws: theral adto thpipes nte ven,ht onduction of he , d ir ovectio n heoven. Cndrng tatte nfunc of convectoisll, we ssume inegigbl o e o tke therml iai ncnuctio ntacctTe heat istranserred r the outideto thensie hile wate in hekeison theontrar。Te temperatue outsideincease more rdly tan tat insid。 An tcotac area twe the pan n thetse k is lrger athabete he pnand the inside cakes,which llusrate why caenthe corner e ovrcoedasiy2.Assumptionsl take e an cakes asbackody, e sorpin of hea in e
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