化工应用数学.ppt

《化工应用数学.ppt》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、化工應用數學授課教師：郭修伯Lecture7PartialDifferentiationandPartialDifferentialEquationsChapter8PartialdifferentiationandP.D.E.sProblemsrequiringthespecificationofmorethanoneindependentvariable.Example,thechangeoftemperaturedistributionwithinasystem:Thedifferentiationp

2、rocesscanbeperformedrelativetoanincrementalchangeinthespacevariablegivingatemperaturegradient,orrateoftemperaturerise.PartialderivativesFigure8.1(contourmapforu)Ifxisallowedtovarywhilstyremainsconstanttheningeneraluwillalsovaryandthederivateofuw.r.t.xwillb

3、etherateofchangeofurelativetox,orthegradientinthechosendirection:uisavectoralongthelineofgreatestslopeandhasamagnitudeequaltothatslope.uwillchangebyduetothechangeinx,andbyduetothechangeiny:Ingeneralform:the“totaldifferential”ofuImportantfactconcerning“par

4、tialderivative”Thesymbol““cannotbecancelledout!Thetwopartsoftheratiodefiningapartialderivativecanneverbeseparatedandconsideredalone.Markedcontrasttoordinaryderivativeswheredx,dycanbetreatedseparatelyChangingtheindependentvariablesw.r.t.uIngeneralform:Indep

5、endentvariablesnottrulyindependentVapourcompositionisafunctionoftemperature,pressureandliquidcomposition:However,boilingtemperatureisafunctionofpressureandliquidcomposision:ThereforeTemperatureincrementofafluid:TotaltimederivativeAspecialcasewhenthepathofa

6、fluidelementistraversedSubstantivederivateofelementoffluidcomparepartialderivateofelementofspaceFormulatingP.D.EsIdentifyindependentvariablesDefinethecontrolvolumeAllowingoneindependentvariabletovaryatatimeApplyrelevantconservationlawUnsteady-stateheatcond

7、uctioninonedimensionLxTxConsideringthethermalequilibriumofasliceofthewallbetweenaplaneatdistancexfromtheheatedsurfaceandaparallelplaneatx+xfromthesamesurfacegivesthefollowingbalance.Rateofheatinputatdistancexandtimet:Rateofheatinputatdistancexandtimet+t

8、:Rateofheatoutputatdistancex+xandtimet:Rateofheatoutputatdistancex+xandtimet+t:HeatcontentoftheelementattimetisHeatcontentoftheelementattimet+tisAccumulationofheatintimetisAverageheatinputduringthetimein