Presentation on theme: "Diffusion Mass Transfer"— Presentation transcript:
1 Diffusion Mass Transfer Chapter 14Sections 14.1 through 14.7
2 General Considerations Mass transfer refers to mass in transit due to a species concentration gradientin a mixture.Must have a mixture of two or more species for mass transfer to occur.The species concentration gradient is the driving potential for transfer.Mass transfer by diffusion is analogous to heat transfer by conduction.Physical Origins of Diffusion:Transfer is due to random molecular motion.Consider two species A and B at the same T and p,but initially separated by a partition.Diffusion in the direction of decreasingconcentration dictates net transport ofA molecules to the right and B moleculesto the left.In time, uniform concentrations of A andB are achieved.
3 Definitions Molar concentration of species i. Mass density (kg/m3) of species i.Molecular weight (kg/kmol) of species i.Molar flux of species i due to diffusion.Transport of i relative to molar average velocity (v*) of mixture.Absolute molar flux of species i.Transport of i relative to a fixed reference frame.Mass flux of species i due to diffusion.Transport of i relative to mass-average velocity (v) of mixture.Absolute mass flux of species i.:in¢¢Transport of i relative to a fixed reference frame.Mole fraction of species iMass fraction of species i
5 Molar and Mass Fluxes of Species A due to Diffusion Diffusion FluxesMolar and Mass Fluxes of Species A due to Diffusionin a Binary Mixture of Species A and BMolar Flux of Species A:By definition:From Fick’s law (mass transfer analog to Fourier’s law):Binary diffusion coefficient or mass diffusivity (m2/s)Mass Flux of Species A:By definition:From Fick’s law:
6 Absolute Molar and Mass Fluxes of Species A Absolute FluxesAbsolute Molar and Mass Fluxes of Species Ain a Binary Mixture of Species A and BMolar Flux of Species A:Mass Flux of Species A:Special Case of Stationary Medium:Achieved to a good approximation for
7 Conservation of Species Application to a Control Volume at an Instant of Time:Application in Cartesian Coordinates to a Differential Control Volume for aStationary Medium of Constant DAB and C orSpecies Diffusion Equation on a Molar Basis:Species Diffusion Equation on a Mass Basis:
8 Conservation of Species (cont) Boundary Conditions (Molar Basis):Consider a Gas (A) / Liquid (B) orGas (A) / Solid (B) Interface.Known surface concentration:For weakly soluble conditions of a gas A in liquid B,(Henry’s law)For gas A in a uniform solid B,Heterogeneous (surface) reactions (Catalysis)
9 Special Cases for One-Dimensional , Steady-State Diffusion in a Stationary MediumDiffusion without Homogeneous Chemical ReactionsFor Cartesian coordinates, the molar form of the species diffusion equation is(1)Plane wall with known surface concentrations:Results for cylindrical and spherical shells Table 14.1
10 Planar medium with a first-order catalytic surface: Special Cases (cont)Planar medium with a first-order catalytic surface:Assuming depletion of species A at the catalytic surface (x = 0),Reaction rate constant (m/s)
11 Process is reaction limited: Special Cases (cont)Assuming knowledge of the concentration at a distance x=L from the surface,Solution to the species diffusion equation (1) yields a linear distribution forHence, at the surface,Limiting Cases:Process is reaction limited:
12 Process is diffusion limited: Special Cases (cont)Process is diffusion limited:Equimolar counterdiffusion:Occurs in an ideal gas mixture if p and T, and hence C, are uniform.
13 Diffusion with Homogeneous Chemical Reactions Special Cases (cont)Diffusion with Homogeneous Chemical ReactionsFor Cartesian coordinates, the molar form of the species diffusion equation isFor a first-order reaction that results in consumption of species A,and the general solution to the diffusion equation isConsider diffusion and homogeneous reaction of gas A in a liquid (B) containerwith an impermeable bottom:
14 Special Cases (cont)Boundary conditionsSolution
15 Evaporation in a Column: A Nonstationary Medium Column EvaporationEvaporation in a Column: A Nonstationary MediumSpecial Features:Evaporation of A from the liquid interfaceInsolubility of species B in the liquid. Hence downward motion by diffusionmust be balanced by upward bulk motion (advection) such that the absoluteflux is everywhere zero.Upward transport of A by diffusion is therefore augmented by advection.
17 One-Dimensional, Transient Diffusion in a Stationary Medium without Homogeneous Chemical ReactionsSpecies Diffusion Equation in Cartesian coordinatesInitial and Boundary Conditions for a Plane Wall with Symmetrical Surface ConditionsNondimensionalizationMass transfer Fourier number
18 Transient Diffusion (cont) Species Diffusion EquationInitial and Boundary ConditionsAnalogous to transient heat transfer by conduction in a plane wall with symmetricalsurface conditions for which and henceHence, the corresponding one-term approximate solution for conduction may beapplied to the diffusion problem by making the substitutionsTable 14.2 summarizes analogy between heat and mass transfer variables.