大学化工应用数学英文课件

DifferentialequationAnequationrelatingadependentvariabletooneormoreindependentvariablesbymeansofitsdifferentialcoefficientswithrespecttotheindependentvariablesiscalleda“differentialequation”.Ordinarydifferentialequation--------onlyoneindependentvariableinvolved:xPartialdifferentialequation---------------morethanoneindependentvariableinvolved:x,y,z,/sundae_mengDifferentialequationAnequati1OrderanddegreeTheorderofadifferentialequationisequaltotheorderofthehighestdifferentialcoefficientthatitcontains.Thedegreeofadifferentialequationisthehighestpowerofthehighestorderdifferentialcoefficientthattheequationcontainsafterithasbeenrationalized.3rdorderO.D.E.1stdegreeO.D.E./sundae_mengOrderanddegreeTheorderofa2Linearornon-linearDifferentialequationsaresaidtobenon-linearifanyproductsexistbetweenthedependentvariableanditsderivatives,orbetweenthederivativesthemselves.Productbetweentwoderivatives----non-linearProductbetweenthedependentvariablethemselves----non-linear/sundae_mengLinearornon-linearDifferenti3FirstorderdifferentialequationsNogeneralmethodofsolutionsof1stO.D.E.sbecauseoftheirdifferentdegreesofcomplexity.Possibletoclassifythemas:exactequationsequationsinwhichthevariablescanbeseparatedhomogenousequationsequationssolvablebyanintegratingfactor/sundae_mengFirstorderdifferentialequat4ExactequationsExact?Generalsolution:F(x,y)=CForexample/sundae_mengExactequationsExact?Generals5Separable-variablesequationsInthemostsimplefirstorderdifferentialequations,theindependentvariableanditsdifferentialcanbeseparatedfromthedependentvariableanditsdifferentialbytheequalitysign,usingnothingmorethanthenormalprocessesofelementaryalgebra.Forexample/sundae_mengSeparable-variablesequationsI6HomogeneousequationsHomogeneous/nearlyhomogeneous?Adifferentialequationofthetype,Suchanequationcanbesolvedbymakingthesubstitutionu=y/xandthereafterintegratingthetransformedequation.istermedahomogeneousdifferentialequationofthefirstorder./sundae_mengHomogeneousequationsHomogeneo7HomogeneousequationexampleLiquidbenzeneistobechlorinatedbatchwisebyspargingchlorinegasintoareactionkettlecontainingthebenzene.Ifthereactorcontainssuchanefficientagitatorthatallthechlorinewhichentersthereactorundergoeschemicalreaction,andonlythehydrogenchloridegasliberatedescapesfromthevessel,estimatehowmuchchlorinemustbeaddedtogivethemaximumyieldofmonochlorbenzene.Thereactionisassumedtotakeplaceisothermallyat55Cwhentheratiosofthespecificreactionrateconstantsare:k1=8k2;k2=30k3C6H6+Cl2

C6H5Cl+HClC6H5Cl+Cl2C6H4Cl2+HClC6H4Cl2+Cl2C6H3Cl3+HCl/sundae_mengHomogeneousequationexampleLi8Takeabasisof1moleofbenzenefedtothereactorandintroducethefollowingvariablestorepresentthestageofsystemattime,p=molesofchlorinepresentq=molesofbenzenepresentr=molesofmonochlorbenzenepresents=molesofdichlorbenzenepresentt=molesoftrichlorbenzenepresentThenq+r+s+t=1andthetotalamountofchlorineconsumedis:y=r+2s+3tFromthematerialbalances:in-out=accumulationu=r/q/sundae_mengTakeabasisof1moleofbenz9EquationssolvedbyintegratingfactorThereexistsafactorbywhichtheequationcanbemultipliedsothattheonesidebecomesacompletedifferentialequation.Thefactoriscalled“theintegratingfactor”.wherePandQarefunctionsofxonlyAssumingtheintegratingfactorRisafunctionofxonly,thenistheintegratingfactor/sundae_mengEquationssolvedbyintegratin10ExampleSolveLetz=1/y3integralfactor/sundae_mengExampleSolveLetz=1/y3integr11Summaryof1stO.D.E.Firstorderlineardifferentialequationsoccasionallyariseinchemicalengineeringproblemsinthefieldofheattransfer,momentumtransferandmasstransfer./sundae_mengSummaryof1stO.D.E.Firstord12FirstO.D.E.inheattransferAnelevatedhorizontalcylindricaltank1mdiameterand2mlongisinsulatedwithasbestoslaggingofthicknessl=4cm,andisemployedasamaturingvesselforabatchchemicalprocess.Liquidat95Cischargedintothetankandallowedtomatureover5days.Ifthedatabelowapplies,calculatedthefinaltemperatureoftheliquidandgiveaplotoftheliquidtemperatureasafunctionoftime.Liquidfilmcoefficientofheattransfer(h1) =150W/m2CThermalconductivityofasbestos(k) =0.2W/mCSurfacecoefficientofheattransferbyconvectionandradiation(h2) =10W/m2CDensityofliquid() =103kg/m3Heatcapacityofliquid(s) =2500J/kgCAtmospherictemperatureattimeofcharging =20CAtmospherictemperature(t) t=10+10cos(/12)Timeinhours()Heatlossthroughsupportsisnegligible.Thethermalcapacityofthelaggingcanbeignored./sundae_mengFirstO.D.E.inheattransfer13TAreaoftank(A)=(x1x2)+2(1/4x12)=2.5m2TwTsRateofheatlossbyliquid=h1A(T-Tw)Rateofheatlossthroughlagging=kA/l(Tw-Ts)Rateofheatlossfromtheexposedsurfaceofthelagging=h2A(Ts-t)tAtsteadystate,thethreeratesareequal:Consideringthethermalequilibriumoftheliquid,inputrate-outputrate=accumulationrateB.C.=0,T=95/sundae_mengTAreaoftank(A)=(x1x214SecondO.D.E.Purpose:reduceto1stO.D.E.Likelytobereducedequations:Non-linearEquationswherethedependentvariabledoesnotoccurexplicitly.Equationswheretheindependentvariabledoesnotoccurexplicitly.Homogeneousequations.LinearThecoefficientsintheequationareconstantThecoefficientsarefunctionsoftheindependentvariable./sundae_mengSecondO.D.E.Purpose:reducet15Non-linear2ndO.D.E.

-EquationswherethedependentvariablesdoesnotoccurexplicitlyTheyaresolvedbydifferentiationfollowedbythepsubstitution.Whenthepsubstitutionismadeinthiscase,thesecondderivativeofyisreplacedbythefirstderivativeofpthuseliminatingycompletelyandproducingafirstO.D.E.inpandx./sundae_mengNon-linear2ndO.D.E.

-Equat16SolveLetandthereforeintegralfactorerrorfunction/sundae_mengSolveLetandthereforeintegral17Non-linear2ndO.D.E.

-EquationswheretheindependentvariablesdoesnotoccurexplicitlyTheyaresolvedbydifferentiationfollowedbythepsubstitution.Whenthepsubstitutionismadeinthiscase,thesecondderivativeofyisreplacedasLet/sundae_mengNon-linear2ndO.D.E.

-Equat18SolveLetandthereforeSeparatingthevariables/sundae_mengSolveLetandthereforeSeparatin19Non-linear2ndO.D.E.-HomogeneousequationsThehomogeneous1stO.D.E.wasintheform:Thecorrespondingdimensionlessgroupcontainingthe2nddifferentialcoefficientisIngeneral,thedimensionlessgroupcontainingthenthcoefficientisThesecondorderhomogenousdifferentialequationcanbeexpressedinaformanalogousto,viz.Assumingu=y/xAssumingx=etIfinthisform,calledhomogeneous2ndODE/sundae_mengNon-linear2ndO.D.E.-Homogen20SolveDividingby2xyhomogeneousLetLetSingularsolutionGeneralsolution/sundae_mengSolveDividingby2xyhomogeneou21Agraphiteelectrode15cmindiameterpassesthroughafurnacewallintoawatercoolerwhichtakestheformofawatersleeve.Thelengthoftheelectrodebetweentheoutsideofthefurnacewallanditsentryintothecoolingjacketis30cm;andasasafetyprecautiontheelectrodeininsulatedthermallyandelectricallyinthissection,sothattheoutsidefurnacetemperatureoftheinsulationdoesnotexceed50C.Ifthelaggingisofuniformthicknessandthemeanoverallcoefficientofheattransferfromtheelectrodetothesurroundingatmosphereistakentobe1.7W/Cm2ofsurfaceofelectrode;andthetemperatureoftheelectrodejustoutsidethefurnaceis1500C,estimatethedutyofthewatercoolerifthetemperatureoftheelectrodeattheentrancetothecooleristobe150C.Thefollowingadditionalinformationisgiven.Surroundingtemperature =20CThermalconductivityofgraphite kT=k0-T=152.6-0.056TW/mCThetemperatureoftheelectrodemaybeassumeduniformatanycross-section.xTT0/sundae_mengAgraphiteelectrode15cmin22xTT0ThesectionalareaoftheelectrodeA=1/4x0.152=0.0177m2Aheatbalanceoverthelengthofelectrodexatdistancexfromthefurnaceisinput-output=accumulationwhere U=overallheattransfercoefficientfromtheelectrodetothesurroundings D=electrodediameter/sundae_mengxTT0Thesectionalareaofthe23Integratingfactor/sundae_mengIntegratingfactorhttp://www.d24LineardifferentialequationsTheyarefrequentlyencounteredinmostchemicalengineeringfieldsofstudy,rangingfromheat,mass,andmomentumtransfertoappliedchemicalreactionkinetics.Thegenerallineardifferentialequationofthenthorderhavingconstantcoefficientsmaybewritten:where(x)isanyfunctionofx./sundae_mengLineardifferentialequationsT252ndorderlineardifferentialequationsThegeneralequationcanbeexpressedintheformwhereP,Q,andRareconstantcoefficientsLetthedependentvariableybereplacedbythesumofthetwonewvariables:y=u+vThereforeIfvisaparticularsolutionoftheoriginaldifferentialequationThegeneralsolutionofthelineardifferentialequationwillbethesumofa“complementaryfunction”anda“particularsolution”.purpose/sundae_meng2ndorderlineardifferential26ThecomplementaryfunctionLetthesolutionassumedtobe:auxiliaryequation(characteristicequation)UnequalrootsEqualrootsRealrootsComplexroots/sundae_mengThecomplementaryfunctionLet27UnequalrootstoauxiliaryequationLettherootsoftheauxiliaryequationbedistinctandofvaluesm1andm2.Therefore,thesolutionsoftheauxiliaryequationare:ThemostgeneralsolutionwillbeIfm1andm2arecomplexitiscustomarytoreplacethecomplexexponentialfunctionswiththeirequivalenttrigonometricforms.

/sundae_mengUnequalrootstoauxiliaryequ28Solveauxiliaryfunction/sundae_mengSolveauxiliaryfunctionhttp://29EqualrootstoauxiliaryequationLettherootsoftheauxiliaryequationequalandofvaluem1=m2=m.Therefore,thesolutionoftheauxiliaryequationis:LetwhereVisafunctionofx/sundae_mengEqualrootstoauxiliaryequat30Solveauxiliaryfunction/sundae_mengSolveauxiliaryfunctionhttp://31Solveauxiliaryfunction/sundae_mengSolveauxiliaryfunctionhttp://32ParticularintegralsTwomethodswillbeintroducedtoobtaintheparticularsolutionofasecondlinearO.D.E.Themethodofundeterminedcoefficientsconfinedtolinearequationswithconstantcoefficientsandparticularformof(x)Themethodofinverseoperatorsgeneralapplicability/sundae_mengParticularintegralsTwomethod33MethodofundeterminedcoefficientsWhen(x)isconstant,sayC,aparticularintegralofequationisWhen(x)isapolynomialoftheformwhereallthecoefficientsareconstants.TheformofaparticularintegralisWhen(x)isoftheformTerx,whereTandrareconstants.Theformofaparticularintegralis/sundae_mengMethodofundeterminedcoeffic34MethodofundeterminedcoefficientsWhen(x)isoftheformGsinnx+Hcosnx,whereGandHareconstants,theformofaparticularsolutionisModifiedprocedurewhenatermintheparticularintegralduplicatesaterminthecomplementaryfunction./sundae_mengMethodofundeterminedcoeffic35SolveEquatingcoefficientsofequalpowersofxauxiliaryequation/sundae_mengSolveEquatingcoefficientsof36MethodofinverseoperatorsSometimes,itisconvenienttorefertothesymbol“D”asthedifferentialoperator:But,/sundae_mengMethodofinverseoperatorsSom37ThedifferentialoperatorDcanbetreatedasanordinaryalgebraicquantitywithcertainlimitations.(1)Thedistributionlaw:A(B+C)=AB+ACwhichappliestothedifferentialoperatorD(2)Thecommutativelaw:AB=BAwhichdoesnotingeneralapplytothedifferentialoperatorDDxyxDy(D+1)(D+2)y=(D+2)(D+1)y(3)Theassociativelaw:(AB)C=A(BC)whichdoesnotingeneralapplytothedifferentialoperatorDD(Dy)=(DD)yD(xy)=(Dx)y+x(Dy)Thebasiclawsofalgebrathusapplytothepureoperators,buttherelativeorderofoperatorsandvariablesmustbemaintained./sundae_mengThedifferentialoperatorDca38DifferentialoperatortoexponentialsMoreconvenient!/sundae_mengDifferentialoperatortoexpon39Differentialoperatortotrigonometricalfunctionswhere“Im”representstheimaginarypartofthefunctionwhichfollowsit./sundae_mengDifferentialoperatortotrigo40TheinverseoperatorTheoperatorDsignifiesdifferentiation,i.e.D-1isthe“inverseoperator”andisan“intergrating”operator.ItcanbetreatedasanalgebraicquantityinexactlythesamemannerasD/sundae_mengTheinverseoperatorTheoperat41Solvedifferentialoperatorbinomialexpansion=2/sundae_mengSolvedifferentialoperatorbino42如果f(p)=0,使用因次分析非0的部分y=1,p=0,即將D-p換為Dintegration/sundae_meng如果f(p)=0,使用因次分析非0的部分y=1,43Solvedifferentialoperatorf(p)=0integrationy=yc

+yp/sundae_mengSolvedifferentialoperatorf(p)44Solvedifferentialoperatorexpandingeachtermbybinomialtheoremy=yc

+yp/sundae_mengSolvedifferentialoperatorexpa45O.D.EinChemicalEngineeringAtubularreactoroflengthLand1m2incrosssectionisemployedtocarryoutafirstorderchemicalreactioninwhichamaterialAisconvertedtoaproductB,Thespecificreactionrateconstantisks-1.Ifthefeedrateisum3/s,thefeedconcentrationofAisCo,andthediffusivityofAisassumedtobeconstantatDm2/s.DeterminetheconcentrationofAasafunctionoflengthalongthereactor.Itisassumedthatthereisnovolumechangeduringthereaction,andthatsteadystateconditionsareestablished.AB/sundae_mengO.D.EinChemicalEngineeringA46uC0xLxuCAmaterialbalancecanbetakenovertheelementoflengthxatadistancexfomtheinletTheconcentraionwillvaryintheentrysectionduetodiffusion,butwillnotvaryinthesectionfollowingthereactor.(WehnerandWilhelm,1956)xx+xBulkflowofADiffusionofAInput-Output+Generation=Accumulation分開兩個sectionCe/sundae_menguC0xLxuCAmaterialbalanceca47dividingbyxrearrangingauxillaryfunctionIntheentrysectionauxillaryfunction/sundae_mengdividingbyxrearrangingauxil48B.C.B.C.ifdiffusionisneglected(D0)/sundae_mengB.C.B.C.ifdiffusionisne49Thecontinuoushydrolysisoftallowinaspraycolumn連續牛油水解1.017kg/sofatallowfatmixedwith0.286kg/sofhighpressurehotwaterisfedintothebaseofaspraycolumnoperatedatatemperature232Candapressureof4.14MN/m2.0.519kg/sofwateratthesametemperatureandpressureissprayedintothetopofthecolumnanddescendsintheformofdropletsthroughtherisingfatphase.Glycerineisgeneratedinthefatphasebythehydrolysisreactionandisextractedbythedescendingwatersothat0.701kg/soffinalextractcontaining12.16%glycerineiswithdrawncontinuouslyfromthecolumnbase.Simultaneously1.121kg/soffattyacidraffinatecontaining0.24%glycerineleavesthetopofthecolumn.Iftheeffectiveheightofthecolumnis2.2mandthediameter0.66m,theglycerineequivalentintheenteringtallow8.53%andthedistributionratioofglycerinebetweenthewaterandthefatphaseatthecolumntemperatureandpressureis10.32,estimatetheconcentrationofglycerineineachphaseasafunctionofcolumnheight.Alsofindoutwhatfractionofthetowerheightisrequiredprincipallyforthechemicalreaction.Thehydrolysisreactionispseudofirstorderandthespecificreactionrateconstantis0.0028s-1.Glycerin,甘油/sundae_mengThecontinuoushydrolysisoft50TallowfatHotwaterGkg/sExtractRaffinateLkg/sLkg/sxHzHx0z0y0Gkg/syHhxzx+xz+zy+yyhx=weightfractionofglycerineinraffinatey=weightfractionofglycerineinextracty*=weightfractionofglycerineinextractinequilibriumwithxz=weightfractionofhydrolysablefatinraffinate/sundae_mengTallowfatHotwaterGkg/sExtr51Considerthechangesoccurringintheelementofcolumnofheighth:Glycerinetransferredfromfattowaterphase,S:sectionalareaoftowera:interfacialareapervolumeoftowerK:overallmasstranstercoefficientRateofdestructionoffatbyhydrolysis,Aglycerinebalanceovertheelementhis:Rateofproductionofglycerinebyhydrolysis,k:specificreactionrateconstant:massoffatperunitvolumeofcolumn(730kg/m3)w:kgfatperkgglycetineAglycerinebalancebetweentheelementandthebaseofthetoweris:Lkg/sxHzHx0z0y0Gkg/syHhxzx+xz+zy+yyhTheglycerineequilibriumbetweenthephasesis:inthefatphaseintheextractphaseinthefatphaseintheextractphase/sundae_mengConsiderthechangesoccurring522ndO.D.E.withconstantcoefficientsComplementaryfunctionParticularsolutionConstantattherighthandside,yp=C/R/sundae_meng2ndO.D.E.withconstantcoeff53B.C.Wedon’treallywantxhere!Applytheequationstwoslidesearlier(replacey*withmx)Wedon’tknowy0,eitherSubstitutey0intermsofothervariables/sundae_mengB.C.Wedon’treallywantxher54/sundae_meng/sundae_me55SimultaneousdifferentialequationsThesearegroupsofdifferentialequationscontainingmorethanonedependentvariablebutonlyoneindependentvariable.Intheseequations,allthederivativesofthedifferentdependentvariablesarewithrespecttotheoneindependentvariable.Ourpurpose:Usealgebraiceleminationofthevariablesuntilonlyonedifferentialequationrelatingtwoofthevariablesremains./sundae_mengSimultaneousdifferentialequa56EliminationofvariableIndependentvariableordependentvariables?Eliminationofindependentvariable較少用EliminationofoneormoredependentvariablesItinvolveswithequationsofhighorderanditwouldbebettertomakeuseofmatricesSolvingdifferentialequationssimultaneouslyusingmatriceswillbeintroducedlaterintheterm/sundae_mengEliminationofvariableIndepen57EliminationofdependentvariablesSolve/sundae_mengEliminationofdependentvaria58=Ey=yc

+yp/sundae_meng=Ey=yc+yphttp://www.docin591.25kg/sofsulphuricacid(heatcapacity1500J/kgC)istobecooledinatwo-stagecounter-currentcoolerofthefollowingtype.Hotacidat174Cisfedtoatankwhereitiswellstirredincontactwithcoolingcoils.Thecontinuousdischargefromthistankat88Cflowstoasecondstirredtankandleavesat45C.Coolingwaterat20Cflowsintothecoilofthesecondtankandthencetothecoilofthefirsttank.Thewaterisat80Casitleavesthecoilofthehotacidtank.Towhattemperatureswouldthecontentsofeachtankriseifduetotroubleinthesupply,thecoolingwatersuddenlystoppedfor1h?Onrestorationofthewatersupply,waterisputonthesystemattherateof1.25kg/s.Calculatetheaciddischargetemperatureafter1h.Thecapacityofeachtankis4500kgofacidandtheoverallcoefficientofheattransferinthehottankis1150W/m2Candinthecoldertank750W/m2C.Theseconstantsmaybeassumedconstant.ExampleofsimultaneousO.D.E.s/sundae_meng1.25kg/sofsulphuricacid(h601.25kg/s0.96kg/s88C45C174C20C40C80CSteadystatecalculation:Heatcapacityofwater4200J/kgCHeattransferareaA1HeattransferareaA2Tank1Tank2Note:和單操課本不同/sundae_meng1.25kg/s0.96kg/s88C45C17461Whenwaterfailsfor1hour,heatbalancefortank1andtank2:Tank1Tank2M:massflowrateofacidC:heatcapacityofacidV:masscapacityoftankTi:temperatureoftankiB.C.t=0,T1=88t=1,T1=142.4Cintegralfactor,ett=1,T2=94.9CB.C.t=0,T2=45/sundae_mengWhenwaterfailsfor1hour,h62Whenwatersupplyrestoresafter1hour,heatbalancefortank1andtank2:Tank1Tank2W:massflowrateofwaterCw:heatcapacityofwatert1:temperatureofwaterleavingtank1t2:temperatureofwaterleavingtank2t3:temperatureofwaterenteringtank212T0T1T2t3t2t1Heattransferrateequationsforthetwotanks:4equationshavetobesolvedsimultaneously/sundae_mengWhenwatersupplyrestoresaft63觀察各dependentvariable出現次數,發現t1

出現次數最少，先消去！(i.e.t1=***代入)再由出現次數次少的

t2

消去…..代入各數值…..B.C.t=0,T2=94.9C同時整理T1/sundae_meng觀察各dependentvariable出現次數,發現64基本上，1storderO.D.E.應該都解的出來，方法不外乎：CheckexactSeparatevariableshomogenousequations,u=y/xequationssolvablebyanintegratingfactor2ndorder以上的O.D.ENon-linearO.D.E.LinearO.D.E.缺x

的O.D.E.,reducedto1stO.D.E.缺y的O.D.E.,reducedto1stO.D.E.homogeneous的O.D.E.,u=y/x我們會解的部分Generalsolution=complementarysolution+particularsolution我們會解的部分constantcoefficientsThemethodofundeterminedcoefficientsThemethodofinverseoperators尋找特殊解的方法variablecoefficient?用數列解，nextcourse/sundae_meng基本上，1storderO.D.E.應該都解的出來，方65DifferentialequationAnequationrelatingadependentvariabletooneormoreindependentvariablesbymeansofitsdifferentialcoefficientswithrespecttotheindependentvariablesiscalleda“differentialequation”.Ordinarydifferentialequation--------onlyoneindependentvariableinvolved:xPartialdifferentialequation---------------morethanoneindependentvariableinvolved:x,y,z,/sundae_mengDifferentialequationAnequati66OrderanddegreeTheorderofadifferentialequationisequaltotheorderofthehighestdifferentialcoefficientthatitcontains.Thedegreeofadifferentialequationisthehighestpowerofthehighestorderdifferentialcoefficientthattheequationcontainsafterithasbeenrationalized.3rdorderO.D.E.1stdegreeO.D.E./sundae_mengOrderanddegreeTheorderofa67Linearornon-linearDifferentialequationsaresaidtobenon-linearifanyproductsexistbetweenthedependentvariableanditsderivatives,orbetweenthederivativesthemselves.Productbetweentwoderivatives----non-linearProductbetweenthedependentvariablethemselves----non-linear/sundae_mengLinearornon-linearDifferenti68FirstorderdifferentialequationsNogeneralmethodofsolutionsof1stO.D.E.sbecauseoftheirdifferentdegreesofcomplexity.Possibletoclassifythemas:exactequationsequationsinwhichthevariablescanbeseparatedhomogenousequationsequationssolvablebyanintegratingfactor/sundae_mengFirstorderdifferentialequat69ExactequationsExact?Generalsolution:F(x,y)=CForexample/sundae_mengExactequationsExact?Generals70Separable-variablesequationsInthemostsimplefirstorderdifferentialequations,theindependentvariableanditsdifferentialcanbeseparatedfromthedependentvariableanditsdifferentialbytheequalitysign,usingnothingmorethanthenormalprocessesofelementaryalgebra.Forexample/sundae_mengSeparable-variablesequationsI71HomogeneousequationsHomogeneous/nearlyhomogeneous?Adifferentialequationofthetype,Suchanequationcanbesolvedbymakingthesubstitutionu=y/xandthereafterintegratingthetransformedequation.istermedahomogeneousdifferentialequationofthefirstorder./sundae_mengHomogeneousequationsHomogeneo72HomogeneousequationexampleLiquidbenzeneistobechlorinatedbatchwisebyspargingchlorinegasintoareactionkettlecontainingthebenzene.Ifthereactorcontainssuchanefficientagitatorthatallthechlorinewhichentersthereactorundergoeschemicalreaction,andonlythehydrogenchloridegasliberatedescapesfromthevessel,estimatehowmuchchlorinemustbeaddedtogivethemaximumyieldofmonochlorbenzene.Thereactionisassumedtotakeplaceisothermallyat55Cwhentheratiosofthespecificreactionrateconstantsare:k1=8k2;k2=30k3C6H6+Cl2

C6H5Cl+HClC6H5Cl+Cl2C6H4Cl2+HClC6H4Cl2+Cl2C6H3Cl3+HCl/sundae_mengHomogeneousequationexampleLi73Takeabasisof1moleofbenzenefedtothereactorandintroducethefollowingvariablestorepresentthestageofsystemattime,p=molesofchlorinepresentq=molesofbenzenepresentr=molesofmonochlorbenzenepresents=molesofdichlorbenzenepresentt=molesoftrichlorbenzenepresentThenq+r+s+t=1andthetotalamountofchlorineconsumedis:y=r+2s+3tFromthematerialbalances:in-out=accumulationu=r/q/sundae_mengTakeabasisof1moleofbenz74EquationssolvedbyintegratingfactorThereexistsafactorbywhichtheequationcanbemultipliedsothattheonesidebecomesacompletedifferentialequation.Thefactoriscalled“theintegratingfactor”.wherePandQarefunctionsofxonlyAssumingtheintegratingfactorRisafunctionofxonly,thenistheintegratingfactor/sundae_mengEquationssolvedbyintegratin75ExampleSolveLetz=1/y3integralfactor/sundae_mengExampleSolveLetz=1/y3integr76Summaryof1stO.D.E.Firstorderlineardifferentialequationsoccasionallyariseinchemicalengineeringproblemsinthefieldofheattransfer,momentumtransferandmasstransfer./sundae_mengSummaryof1stO.D.E.Firstord77FirstO.D.E.inheattransferAnelevatedhorizontalcylindricaltank1mdiameterand2mlongisinsulatedwithasbestoslaggingofthicknessl=4cm,andisemployedasamaturingvesselforabatchchemicalprocess.Liquidat95Cischargedintothetankandallowedtomatureover5days.Ifthedatabelowapplies,calculatedthefinaltemperatureoftheliquidandgiveaplotoftheliquidtemperatureasafunctionoftime.Liquidfilmcoefficientofheattransfer(h1) =150W/m2CThermalconductivityofasbestos(k) =0.2W/mCSurfacecoefficientofheattransferbyconvectionandradiation(h2) =10W/m