1、高等化工传递过程原理（研究生）高等化工传递过程原理（研究生）全册配套完整课件全册配套完整课件3Basic ConceptsFluxDriving Force Constitutive Equation Transport PropertyFlux Bulk Movement Convective transport Convective Flux Small scale molecular displacement Molecular(or Diffusive) transport Molecular (or Diffusive) Flux Flux= Convective Flux + Mo
2、lecular Flux Flux=(“concentration”) (“transport velocity”)abvaatbbFvfvaftbvFaatbfvviiiiiCnNvthe molar flux of species irelative to fixed coordinatesthe mass flux of species irelative to fixed coordinatesthe velocity of species irelative to fixed coordinatesthe mole concentration of species ithe mass
3、 concentration of species i1iiintiiCCxCCTransport of Chemical Species mass fraction mole fraction1iiintiiBulk velocity or reference velocity of a mixtureSpecies velocity relative to the mixture: diffusive velocity111nnnaMiiiiiiiiiaxvvvvvvaiv - viv - vMiv - vaaiiiCJv - vaaiiiCNvJaaiiinvjaaiiijv - v11
4、niiaSpecies Flux Reference velocity Molar units Mass units 0 Ni ni v Ji ji vM JiM jiM1111,0,0nnMMMMiiiiiiiitiiinnMMiiiiiiiitiiiCCCCNvJvJvNvJnvjvjvnvjFlux relationships1vnaiitiian10naiiiiaj1J0naiiiiax1vnaiitiiaCxNTransformations between Species Fluxes( )()MnN()()aaMjJ ,1,2,1i kn,1,2,1i kn aabbAjjabkn
5、kikikiknkaa bAbaabbBJJabknkikikiknkaa bBxxb xGradientEnergy flux Flux of species iTemperatureConcentration of specie iConcentration of species jElectrical potentialPressureOther external forcesConduction (Fourier)Diffusion-thermo effect (Dufour)Thermal diffusion(Soret)Binary Diffusion (Fick)Multicom
6、ponent diffusion (Maxwell-Stefan)Ion migration (M-S)Pressure diffusion (M-S)M-SMolecular Flux = f (Transport Properties, Driving forces)Driving forces = GradientsDominant gradient : Primary transport propertyOther gradients : Secondary transport properties Constitutive Equations for Molecular Transp
7、ort Linear Flux-Gradient Laws (Classic Constitutive Equations) :Molecular Flux = -Transport PropertyGradientHeat Transport-Fouriers Law of Conduction Momentum Transport- Newtons Law of ViscosityMass TransportFicks Law of DiffusionFicks law for binary mixtures of A and B tABtABAAAAMMAtADDMC MAReferen
8、ce Velocity Mass units Molar units vj- J- vj-MABAAtABADxC Dx J-Diffusion of Chemical SpeciesMagnitudes of Transport CoefficientsBinary Diffusivity (Ficks )10-1410-1210-1010-810-610-410-2GasesLiquidsBinary Diffusivity (m2 s-1)SolidsyTkqTkqyyynqeennqn if)(nqxyzn=eyqTkqHeat ConductionT qFor isotropic m
9、aterialsFor anisotropic materialsMagnitudes of Transport CoefficientsThermal conductivity10-310-210-1100101102103GasesNonmetallicLiquidsThermal Conductivity (W m-1 K-1)LiquidMetalsNonmetallicSolidsSolidMetalsStress and Momentum Fluxxyxdvdy t vvns(n)xyzn=eysns1s2s(n)yzyxyy( ) ( )()xxxyxzyxyyyzzxzyzzy
10、yxxyyyyzzs nn s ns eeee xxyvev Magnitudes of Transport CoefficientsViscosity 10-610-510-410-310-210-1100GasesLiquidsViscosity (Pas)Transport coefficients (Diffusivities) Classic constitutive equations:Molecular Flux = -“Diffusivity” “Concentration Gradient”()yppkqC TyC AAyABCJDy xyxdvdy Prandtl numb
11、er Schmidt number Lewis numberPrpCkABABScDDPrABScLeDConvection relative to Diffusion or conduction Peclet number PeA measure of convective transport relative to molecular-based transportABLLPeorDvvMechanism of Molecular Transport Limitation on Length and Time ScalesLinear Flux-Gradient Laws (Classic
12、 Constitutive Equations) for Molecular Transport:()yppkqC TyC AAyABCJDy -xyxvy Flux = -(Transport property)(Gradient)Theoretical understanding of transport phenomena in molecular basisThe molecular transport of energy, species and momentum are based ultimately on the random motions of molecules.Mode
13、ls: to describe random motions of molecules, and relate the molecular motions to transport fluxesLattice Model x,y,zx,y,z+lx,y,z-lx,y+l,zx,y-l,zx+l,y,zx-l,y,z),(),(6zlyxbzyxbufy),(222),(2),(),(zyxzyxyblyblzyxbzlyxb266ybfDyullD According to elementary kinetic theory for low density gases:Application
14、of Lattice Model to Gases1/228mean molecular velocity mean free path 2: molecular diameter, : Avogadros numberAvAvRTuvMRTld NPdNAvAvABAvPPNdMRTDPMNdRTDDNdCMRTDCk22/12/32/122/32/322/12/3)(31)(31)(312112B andA of mixture afor 1BABAdddMMM1/27/43/2but k (rather than T) (rather than T)ABTDT、The Kinetic T
15、heory of GasesReference:Sears, F.W., An Introduction to Thermodynamics, The Kinetic Theory of Gases, and Statistical Mechanics, 2nd ed., Addison-Wesley, Reading, MA. 1953Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B., The Molecular Theory of Gases and Liquids, J. Wiley & Sons, New York, 1954Chapm
16、an Enskog TheoryLennard-Jones(6-12) potential:03()0.77( )0.841CCbKTAVkGases03()1.15( )1.166bbbKTAVk03si()1.95( )1.222fusionfuonbKTAVkLiquidsSolids( )drdr FChapman, S. and Cowling, T.G., The Mathematical Theory of Non-Uniform Gases, 3rd ed., University Press, Cambridge, 19701264rr Chapman Enskog Theo
17、ry and Gas Viscosity622.669 10MTPa s*bTk T*0.14874*1.161450.524872.16178exp 0.77320exp 2.43787TTT Chapman Enskog Theory and Gas Thermal Conductivity and Diffusivity22/8.332 10/kTMkWmK3722111.86 10/ABABDTMMDmsP2AB*0.14874*1.161450.524872.16178=exp 0.77320exp 2.43787kTTT *0.15610*1.60360.193001.035871
18、.76474exp 0.47635exp 1.52996exp 3.89411DTTTT*/bABbABTk Tk T *bTk TStokes-Einstein ModelDiffusion in Liquids is the drag coefficientbAAyByAkT Cf vvCyfAyAAyyJCvvAAyABbABCJDyk TDf 06ABbAk TDr-2/3DHiss, T. G. and Cussler, E. L., “Diffusion in High Viscosity Liquid”, AIChE J., 19, 698 (1973)Eyrings rate
19、theoryFree VolumeGlasstone, S., Laidler, K.J., and Eyring, H., Theory of Rate Processes, McGraw-Hill, New York (1941)Eyring, H., “Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates”, J. Chem. Phys., 4, 283 (1936)Diffusion in Liquids2expbABbk TGDlhk T012expcfbGlk T012expcrbGl
20、k T02exp2cnetbbGlk Tk T 20expAAynetAbGCJlClk Ty expavbN hGVk T2bABavk TlDV N2bABAk TDr0bk ThTransport properties data source1. J. Millat, J.H. Dymond, C.A. Nieto de Castro,“Transport Properties of Fluids: Their Correlation, Prediction and Estimation”, New York, IUPAC, Cambridge University Press,1996
21、2. Carl L. Yaws, “Handbook of Transport Properties Data: Viscosity, Thermal Conductivity, and Diffusion Coefficients of Liquids and Gases”, Houston, Tex, Gulf Pub. Co., 19953. R.C. Reid, J.M. Prausnitz, B.E. Poling, “The Properties of Gases and Liquids” 4th ed., McGraw-Hill, New York, 1987Lattice Mo
22、del for Diffusivities in SolidsSubstitutional diffusionInterstitional diffusionInterstitional diffusionSubstitutional diffusionElementary jump-rate theorylattice sitesfree energysaddle point/u l*GHT S 2*exp()exp()6bblGGDk Tk TGHT S *0exp()bHHDDk T1/3min10/LNnminLL1/3minmin/LNn()L l gases10 ()Ll gase
23、s()L d liquids10 ()Ld liquidsLimitation on Length ScaleEstimates of minimum system dimensions for continuum transport models using bulk propertiesBased on an ideal gas at P=1atm and T=293K, with M=30g/molBased on density=1000kg/m3, and M=30g/molLimitation on Length ScaleLimitation on Time Scale1010(
24、)ts gases1310()ts liquidsConservation Equations for Heat and Mass TransferTransport Model: Equations governing the transport phenomena Three major parts: Constitutive equations Conservation equations for interior points Conservation equations for boundary points General Forms of Conservation Equatio
25、nsControl volume for formulating a desired balance equationvvnnvsvsV(t)S(t)A unit vector n defined at every point on the surface is normal to the surface and directed outward. Conservation Equations for Finite VolumesA fixed control volume: , V=constant, S=constantA moving control volume: , V=V(t),
26、S=S(t)VVSVdVBdSbdVdtdnFVVSVdVBdSdVtbnF( )( )( )( )sVV tS tS tV tdbdVdSbdSB dVdt F nvn( )( )( )(sV tV tS tdbbdVdVbdSdttvn)The Leibniz formula for differentiating a volume integral0sv0svControl volume enclosing part of an interfacevsnAvIVA(t)SB(t)nBnIPhase APhase BSA(t)VB(t)SI(t)()()()()()()()()()()()
27、(tSBAtSstVtVtSBtSstVtVtSAtSstVtVIIBBBIAAAdSbbdSbdVtbbdVdtddSbdSbdVtbbdVdtddSbdSbdVtbbdVdtdIIIIBIIAn)vnvnvnvnvnv( )( )( )( )( )(IIABVSV tStS tV tStbdVbbdSdSB dVB dSt II)vnF nFor the possibility of a source term at an internal interface, if the rate of formation per unit area of interface is donated a
28、s Bs(r,t) ( )( )( )( )( )IsVSV tS tS tV tStdbdVdSbdSB dVB dSdt F nvnConservation Equations for Points inside a Phase-The differential form of the general conservation equationVSdVdSFnF0VVVbB dVtbBt FFThe surface integral is converted to a volume integral by the application of the divergence theorem.
29、VVSVbdVdSB dVt F nDifferential Operations in Rectangular CoordinatesGradient for a differentiable scalar function Divergence for a differentiable vector function Laplacian for a differentiable scalar function fffffxyz xyzeeevyxzvvvxyzvf2222222ffffxyzxxyyzzvvvveeexyz xyzeeeDifferential Operations in
30、Cylindrical CoordinatesGradient for a differentiable scalar function Divergence for a differentiable vector function Laplacian for a differentiable scalar function f1ffffrrz rzeeev11zrvvrvr rrzvf22222211ffffrr rrrzrrzzvvvveeeDifferential Operations in Spherical CoordinatesGradient for a differentiab
31、le scalar function Divergence for a differentiable vector function Laplacian for a differentiable scalar function f11sinffffrrr reeev22111sinsinsinzrvr vvrrrrvf22222222111sinsinsinffffrrrrrrrrvvvveeePoint Conservation Equations for Interfaces-The boundary conditionsSABVVBbbdVBtbIIInvFvFF)()(0Control
32、 volume enclosing part of an interface between phase A and phase B. The Surface SA and SB are each a constant distance l from the interface.( )( )( )( )( )(IIABVSV tStS tV tStbdVbbdSdSB dVB dSt II)vnF n( )( )IISBAStStbbdSB dSIIIF- vF- vnConvective and Diffusive FluxesaaaabbFvfv : convective contribu
33、tionf : diffusive contribution()aaVbbBt vf()() aaaaBASbbBIIIfvvfvvnThe general conservation equation at an interior point is rewritten as:The balance at an interface isSummary for General Conservation EquationsGeneral conservation equations for interior points and interfaces interior points points a
34、t interfacesVBtbF()VbbBt vf()() BASbbBIIIFvFvnSABBbbIIInvvfvvf)()()aaVbbBt vf()() aaaaBASbbBIIIfvvfvvnWith the mass average velocityFlux Continuity and Symmetry Conditions within a Given Phase01lim0SSdSSF nF is a continuous function of position at all interior point. 0nFsymmetry plane00ratFraxisymme
35、tric or spherically symmetric0anf00arfatrTotal mass conservation at interior pointsContinuity Equation: application of the general conservation equations to total mass,0,0()0tVbBt Fnv fvif constant0vat steady state0vContinuity Equationtion0 xyzvvvtxyz11()0rzr vvvtr rrz22111()(sin )0sinsinrrvvvtrrrr
36、Rectangular: Cylindrical: Spherical: Total mass conservation at interface,0,0sbB f1122nInnInvvvvAlternative Conservation EquationsvtDtD/bBVBDtBDfconstant VBDtDbf()bclassic constitutive equation fD2 ,constantVDbbBDtDDMaterial derivative (or substantial derivative): if the amount of any quantity per u
37、nit mass is denoted as Conservation of Heat,pVVpVbC TBHD C THDt fq,qk T q22 constnat ,PVPVPDTCkTHCkDtHDTTDtC For solids or pure fluidsConservation of Heat),(tzyxTT 222222VxyzPHTTTTTTTvvvtxyzxyzC),(tzrTT2222211()VrzPvHTTTTTTTvvrtrrzrrrrzC),(trTT2222222111()(sin)sinsinsinVrPvvHTTTTTTTvrtrrrrrrrrC Rect
38、angular: Cylindrical: Spherical: Heat Transfer at Interfaces1. Interfacial Energy Balance12,SSnnInBHvvvfq21( , )( , )( , )nnSqtqtHtsssrrr0),(tHSsrfor the usual situation Heat Transfer at Interfaces2. Thermal Equilibrium at an interface Usually assumption: Certain situations:3. Symmetry Conditions12T
39、T12TT0nq00ratqrsymmetry planeaxisymmetric or spherically symmetricConservation of Chemical SpeciesiiViCRt N2iiiViDCDCRDtConstant and iDWith Ficks Law for iJViViiRBCb,NFiiiCNv+JViiiiiiziyixiRzCyCxCDzCvyCvxCvtC2222222222211()iiiiiiirziViCCvCCCCCvvDrRtrrzrrrrz2222222111()(sin)sinsinsiniiiiiiiriVivCCvCC
40、CCCvDrRtrrrrrrrrConservation of Chemical SpeciesRectangular: Cylindrical: Spherical: Mass Transfer at Interfaces 221121212212( , )( , )( , )( , )( , )( , );0 (phase 1 is an impermeable) ; (phase 1 is an impermeable)iniIniniInSiininSiIninSiininSiInnninSiNtCvNtCvRtNtNtRtvNRJJRvvvJRssssssrrrrrr12: part
41、ition coefficientiiiiCK CKsymmetry plane00ininNJaxisymmetric or spherically symmetric0000irirNatrJatriiViCRt N121()() iIiiIISiCCRNvNvnTransport of Chemical SpeciesFluxes of Chemical SpeciesReference velocity Molar units Mass units 0 Ni ni v Ji ji vM JiM jiM1111,0,0nnMMMMiiiiiiiiinnMMiiiiiiiiiCCCNvJv
42、JNvJnvjvjnvjFlux relationshipsDiffusion of Chemical Species ABABAAAAMMAAABDDMCM DAReference Velocity Mass units Molar units- vj- J vj-MAAABAxCDx J-Ficks law for binary mixtures of A and BOne-Dimensional Steady ProblemsExample 1: Directional Solidification of a Dilute Binary Alloy022dydCUdyCdDiii(1)(
43、0)(0)()iiiiiiidCUKCdyDCC(1)exp()( )(1)exp()iiiiiiiiiiiiyKKPeC yCKKPeUPeD(1)exp()iSiiiiiCKCKKPe 00iSiSiiCyCK CExample 2: Diffusion in a binary gas with a heterogeneous reaction y=0y=LA, BCA=CA0AmBGasCatalyticsurfacenAsnSACkRdydNdydNByAy 0AyBymNN()(1)AAyAAyBytABAAyAAytABdxNxNNC DdydxNx NmC Ddy1(/)(1)A
44、yAAtABNdCCCmdyD ( )nAySAsnANRk C L 00AACC1 11 )1 (1)1 (1ln)1 (1000mmmxmxmxDaAAAn01(1) (0)1nAdDaxmd 1000nAAsnAAABACLCyk CLDaCLDC0( ) 1(/)(1) (0)nAsnAAtAAABdCkCLCCmCCdyD Example 3: Diffusion in a dilute liquid solution with a reversible homogeneous reaction y=0y=LCA=CA0,CB=CB0ABLiquidInertsurface11111
45、()/VABABAVBRk Ck Ck KCCRKkk 212212()0()0AASBABBSBAd CDk KCCdyd CDk KCCdy00,at 00 at AABBABCCCCydCdCyLdydy222210ABASBSABASBSd Cd CDDdydydCdCDDadydy10, at ayL2ASABSBD CD Ca200 at 0ASABSBaD CD Cy00()ASBAABBSDCCCCD00ABABAAyCKCLCC22221100()()AABASBSAddk Lk LKCDDC )sinh()tanh()cosh()1 ()sinh()tanh()cosh()
46、1 (BA 1010AAdd(1)(1)1ABABeee 0e 00/1/1/BAABABABCCDDKKDD 011ABExample 4: Transient diffusion in a solid from a surface fixed concentrationDoping of semiconductors22CCDtyI.C. C=0 , y 0, t=0B.C. C=Cs, y=0 C=0, y Transient Problems22ffDty,000,1,0fyftft/SfCC yt ffSimilarity method (combination of variabl
47、es)220d fdfdDdcD22cD 00 1/22tcD t2ycDt220d fdfcdd 01;0ff 22cysDt2220d fdfsdsdsThe Error Function 202(0)0()1znerfzednerferf ,12syCy tCerfDt 20211snfsednerfs 22001snnednfsedn Doping of semiconductorsA 2mm thick silicon wafer is to be doped with antimony (Sb) in order to create a p-type region. This ca
48、n be done by passing a SbCl3/H2 gas mixture over the surface of the wafer at 1200 oC, which fixes the surface Sb concentration at 1023 atoms /m3. Suppose that the donor density (which is just another term for the Sb concentration) is hoped to be greater than or equal to 31022 /m3, over a depth of 1m
49、 below the surface. Determine how long the wafer should be exposed to this atmosphere in order to achieve this. 1210.720.732ssyCCerfDtyCerfCDtyDt 0-3172264exp()383.0001.3 10 exp()8.314 1473.23.4 10/1101.38 103.820.73QDDRTmstshDExample 5: Transient diffusion in a Symmetric SlabCA*CA*2Lz=0 zCA=CA0 at
50、t=022AAACCDtz00,AACzC*,00,AAACtCt LCz*0*AAAACCCC22zDt0,1z,00,0tt zz)()(zZtTTZDTZ 0 ZZ 0(0)0Z LZ0 )cos()sin(zBzAZ2(21)2nnL)cos(zBZnnn0 DTT)exp(DtATnnn000cos()exp()nnnnnnnnnT ZCzDt4( 1)(21)nnCn)2) 12(exp()2) 12(cos() 12() 1(420DtLnzLnnnn)2) 12(exp()2) 12(cos() 12() 1(4)(20*0*DtLnzLnnCCCCnnAAAAiiViCRt
