化工应用数学

化工應用數學授課教師：郭修伯助理教授Lecture3應用數學方程式表達物理現象建立數學模式Theconservationlaws–materialbalance–heatbalance–enerybalanceRateequations–therelationshipbetweenflowrateanddrivingforceinthefieldoffluidflow–heattransfer–diffusionofmatter建立數學模式Theconservationlaws–materialbalance–heatbalance–enerybalance(rateof)input-(rateof)output=(rateof)accumulation範例說明Asingle-stagemixersettleristobeusedforthecontinuousextractionofbenzoicacidfromtoluene,usingwaterastheextractingsolvent.ThetwostreamsarefedintoatankAwheretheyarestirredvigorously,andthemixtureisthenpumpedintotankBwhereitisallowedtosettleintotwolayers.Theuppertoluenelayerandthelowerwaterlayerareremovedseparately,andtheproblemistofindwhatproportionofthebenzoicacidhaspassedintothesolventphase.watertoluene+benzoicacidtoluene+benzoicacidwater+benzoicacid簡化（理想化）Sm3/stolueneykg/m3benzoicacidRm3/stoluenexkg/m3benzoicacidSm3/swaterRm3/stolueneckg/m3benzoicacidRateequationfortheextractionefficiency:y=mxMaterialBalance:Inputofbenzoicacid=outputofbenzoicacidRc=Rx+SySamemethodcanbeappliedtomulti-stages.隨時間變化Funtionoftime非穩定狀態(unsteadystate)Inunsteadystateproblems,timeentersasavariableandsomepropertiesofthesystembecomefunctionsoftime.Similartothepreviousexample,butnowassumingthatthemixerissoefficientthatthecompositionsofthetwoliquidstreamsareinequilibriumatalltimes.Astreamleavingthestageisofthesamecompositionasthatphaseinthestage.Thestateofthesystematageneraltimet,wherxandyarenowfunctionsoftime.Sm3/stolueneykg/m3benzoicacidRm3/stoluenexkg/m3benzoicacidSm3/swaterRm3/stolueneckg/m3benzoicacidV1,xV2,yMaterialbalanceonbenzoicacidSm3/stolueneykg/m3benzoicacidRm3/stoluenexkg/m3benzoicacidSm3/swaterRm3/stolueneckg/m3benzoicacidV1,xV2,yInput-output=accumulationdtdxVdtdxVSyRxRc21)(單位時間的變化CmVVtmSRxmSRRc21])(ln[t=0,x=0tmVVmSRmSRRcx21exp1MathematicalModelsSaltaccumulationinastirredtankt=0Tankcontains2m3ofwaterQ:Determinethesaltconcentrationinthetankwhenthetankcontains4m3ofbrineBrineconcentration20kg/m3feedrate0.02m3/sFlow0.01m3/s建立數學模式VandxarefunctionoftimetDuringt:–balanceofbrine–balanceofsaltBrineconcentration20kg/m3feedrate0.02m3/sBrine0.01m3/sVm3xkg/m3tdtdVtt01.002.0VxtdtdxxtdtdVVtxt))((01.02002.0解數學方程式Solvex=20-20(1+0.005t)-2V=2+0.01t0)0(2)0(01.04.001.0xVxdtdxVdtdVxdtdVMathematicalModelsMixingPurewater3l/minMixture2l/minMixture3l/minMixture4l/minMixture1l/minTank1Tank2t=0Tank1contains150gofchlorinedissolvedin20lwaterTank2contains50gofchlorinedissolvedin10lwaterQ:Determinetheamountofchlorineineachtankatanytimet0建立數學模式Letxi(t)representsthenumberofgramsofchlorineintankiattimet.Tank1:x1’(t)=(ratein)-(rateout)Tank2:x2’(t)=(ratein)-(rateout)Mathematicalmodel:x1’(t)=3*0+3*x2/10-2*x1/20-4*x1/20Purewater3l/minMixture2l/minMixture3l/minMixture4l/minMixture1l/minTank1Tank2x2’(t)=4*x1/20-3*x2/10-1*x2/1050)0(150)0(525110310321212211xxxxdtdxxxdtdx解數學方程式Howtosolve?UsingMatricesX’=AX;X(0)=X0where–x1(t)=120e-t/10+30e-3t/5–x2(t)=80e-t/10-30e-3t/55015052511031030XandA50)0(150)0(525110310321212211xxxxdtdxxxdtdxMathematicalModelsMass-SpringSystem–Supposethattheupperweightispulleddownoneunitandthelowerweightisraisedoneunit,thenbothweightsarereleasedfromrestsimultaneouslyattimet=0.Q:Determinethepositionsoftheweightsrelativetotheirequilibruimpositionsatanytimet0k1=6k3=3k2=2m1=1m2=1y2y1建立數學模式Equationofmotionweight1:weight2:Mathematicalmodel:m1y1”(t)=-k1y1+k2(y2-y1)0)0()0(1)0(1)0(52282121212211yyyyyyyyyyk1=6k3=3k2=2m1=1m2=1y2y1m2y2”(t)=-k2(y2-y1)-k3y2解數學方程式Howtosolve?–y1(t)=-1/5cos(2t)+6/5cos(3t)–y2(t)=-2/5cos(2t)-3/5cos(3t)0)0()0(1)0(1)0(52282121212211yyyyyyyyyy隨位置變化FunciotnofpositionMathematicalModelsRadialheattransferthroughacylindricalconductorTemperatureataisToTemperatureatbisT1Q:Determinethetemperaturedistributionasafunctionofratsteadystaterr+drab建立數學模式ConsideringtheelementwiththicknessrAssumingtheheatflowrateperunitarea=QRadialheatfluxAhomogeneoussecondorderO.D.E.))((22rdrdQQrrrQdrdTkQwherekisthethermalconductivity022drdTdrTdr解數學方程式Solve1022)()(0TbTTaTdrdTdrTdr)lnlnlnln)(()(010abarTTTrT流場(Flowsystems)-EulerianTheanalysisofaflowsystemmayproceedfromeitheroftwodifferentpointsofview:–Eulerianmethodtheanalysttakesapositionfixedinspaceandasmallvolumeelementlikewisefixedinspacethelawsofconservationofmass,energy,etc.,areappliedtothisstationarysystemInasteady-statecondition:–theobjectoftheanalysisistodeterminethepropertiesofthefluidasafunctionofposition.流場(Flowsystems)-Lagrangian–theanalysttakesapositionastrideasmallvolumeelementwhichmoveswiththefluid.–Inasteadystatecondition:theobjectiveoftheanalysisistodeterminethepropertiesofthefluidcomprisingthemovingvolumeelementasafunctionoftimewhichhaselapsedsincethevolumeelementfirstenteredthesystem.Thepropertiesofthefluidaredeterminedsolelybytheelapsedtime(i.e.thedifferencebetweentheabsolutetimeatwhichtheelementisexaminedandtheabsolutetimeatwhichtheelemententeredthesystem).–Inasteadystatecondition:boththeelapsedti