18-660:NumericalMethodsforEngineeringDesignandOptimizationXinLiDepartmentofECECarnegieMellonUniversityPittsburgh,PA15213Slide1Overview„Lecture2:OrdinaryDifferentialEquation(ODE)^Numericalintegration^Stability„Lecture3:PartialDifferentialEquation(PDE)^Heatequation^BoundaryconditionSlide2PartialDifferentialEquation(PDE)„Partialdifferentialequationisoftenusedtodescribeaphysicalsystemorprocess„Example:heatequationisanPDEforthermalanalysis^WewillderiveheatequationstepbystepinthislectureSlide3HeatEquation„Heatequationcanbederivedfromseveralfundamentalphysicslaws^Fourier'slawslaw^Conservationofheat^Etc.„Wewillusea1-Dexampletoillustrateheatequation^HelptogetmanyinsightsaboutheattransferprocessSlide41-DHeatModel„Weconsidera1-DrodoflengthL0LΔxx„Threemajorassumptions^Rodismadeofasinglehomogenousconductingmaterial^Rodislaterallyinsulated(heatflowsonlyinthex-direction)^Rodisthin(constanttemperatureatallpointsofacrosssection)Slide5ConservationofHeat„Applyconservationofheattothesegment[x,x+Δx]0LΔxxNetchangeofheatNetfluxofheatTotalheatgeneratedinside[x,x+Δx]acrossboundariesinside[x,x+Δx]WewilllookateachofthesethreecomponentsindetailSlide6ConservationofHeatNetchangeofheatNetfluxofheatTotalheatgeneratedinside[x,x+Δx]acrossboundariesinside[x,x+Δx]xx+Δx„Totalheatinside[x,x+Δx]isequalto:Slide7ConservationofHeatNetchangeofheatNetfluxofheatTotalheatgeneratedinside[x,x+Δx]acrossboundariesinside[x,x+Δx]xx+Δx„Netchangeofheatisequalto:Slide8ConservationofHeatNetchangeofheatNetfluxofheatTotalheatgeneratedinside[x,x+Δx]acrossboundariesinside[x,x+Δx]xx+Δx„Fourier'slaw:heatfluxacrossaboundaryisproportionaltothetemppgeraturegradientacrosstheboundary:xx+ΔxSlide9ConservationofHeatNetchangeofheatNetfluxofheatTotalheatgeneratedinside[x,x+Δx]acrossboundariesinside[x,x+Δx]xx+Δx„Netfluxofheatacrossboundariesisequalto:∂T(x,t)∂T(x+Δx,t)κ⋅A⋅κ⋅A⋅∂x∂xHeatfluxatxxx+ΔxHeatfluxatx+