Presentation on theme: "Diffusion Mass Transfer"— Presentation transcript:
1 Diffusion Mass Transfer Chapter 14Sections 14.1 through 14.7Lecture 18
2 Physical Origins and Rate Equations Mass Transfer in Nonstationary MediaConservation Equation and Diffusion through Stationary MediaDiffusion and Concentrations at InterfacesDiffusion with Homogenous ReactionsTransient Diffusion
3 1. Physical Origins and Rate Equations Driving potential for mass transferConcentration GradientModes of mass transferConvection and Diffusion
4 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.
5 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.Transport of i relative to a fixed reference frame.Mole fraction of species iMass fraction of species i
6 Mixture Composition Definitions: Mass density of species i: i=Mi*Ci (kg/m3)Molecular weight of species i: Mi (kg/kmol)Molar concentration of species i: Ci (kmol/m3)Mixture mass density: (kg/m3)Total number of moles per unit volume of mixture
7 Mixture Composition Definitions: Mass fraction of species i: mi=i/ Molar fraction of species i: xi=Ci/CFor ideal gases:
8 Fick’s Law of Diffusion For transfer of species A in a binary mixture of A & BMass flux of species A (kg/m2s):Molar flux of species A (mol/m2s):Binary diffusivity DAB (m2/s)Ordinary diffusion due to concentration gradient and relative to coordinates that move with average velocity
9 Fick’s Law of Diffusion For transfer of species A in a binary mixture of A & BMass flux of species A (kg/m2s):Molar flux of species A (mol/m2s):If C and are constants, the above equations become:
10 Mass Diffusivity For ideal gases Gas-Gas Liquid-Liquid Gas in Solid At 298 K, 1 atmGas-GasLiquid-LiquidGas in Solid~10-5 m2/s~10-9 m2/s~ m2/sSolid in Solid~ m2/s
11 Example 14.1Consider the diffusion of hydrogen (species A) in air, liquid water, or iron (species B) at T = 293 K. Calculate the species flux on both molar and mass bases if the concentration gradient at a particular location is dCA/dx= 1 kmol/m3●m. Compare the value of the mass diffusivity to the thermal diffusivity. The mole fraction of the hydrogen xA, is much less than unity.
12 Example 14.1Known: Concentration gradient of hydrogen in air, liquid water, or iron at 293 KFind: Molar and mass fluxes of hydrogen and the relative values of the mass diffusivity and thermal diffusivitySchematic:
13 Example 14.1 Assumptions: Steady-state conditions Properties: Table A.8, hydrogen-air (298 K): DAB= 0.41x10-4 m2/s, hydrogen-water (298 K): DAB= 0.63x10-8 m2/s, hydrogen-iron (293 K): DAB= 0.26x10-12 m2/s. Table A.4, air (293 K): α= 21.6x10-6 m2/s; Table A.6, water (293 K): k = W/m•K, ρ = 998 kg/m3, cp= 4182 J/kg K. Table A.l, iron (300 K): α = 23.1 x 10-6m2/s.Analysis: Using Eqn , we can find that the mass diffusivity of hydrogen in air at T=293K is
14 Example 14.1For the case where hydrogen is a dilute species, that is xA<<1, the thermal properties of the medium can be taken to be those of the host medium consisting of species B. The thermal diffusivity of water is:The ratio of the thermal diffusivity to the mass diffusivity is the Lewis number Le, defined in Equation 6.50.The molar flux of hydrogen is described by Fick’s law, Equation
15 Example 14.1 Hence, for the hydrogen-air mixture, The mass flux of hydrogen in air is found to from the expression:
16 Example 14.1The results for the three different mixtures are summarized in the following table:
17 Mass Transfer in Nonstationary Media Absolute Mass FluxFor mass flux relative to a fixed coordinate systemMass flux of species A (kg/m2s):Mass flux of species B (kg/m2s):Mass flux of mixture (kg/m2s):Mass-average velocity (m/s):
18 Relative Mass FluxMass flux of species A (kg/m2s):
19 Relative Mass Flux Mass flux of species A (kg/m2s): For binary mixture of A & B:
20 Relative Mass FluxFor binary mixture of A & B:DAB=DBA
21 Absolute Molar FluxFor molar flux relative to a fixed coordinate system Molar flux of species A (mol/m2s): Mass flux of species B (kg/m2s): Mass flux of mixture (kg/m2s):Molar-average velocity (m/s):
22 Absolute Molar Flux Molar flux of species A (kg/m2s): Mass flux of species A (kg/m2s):
23 Absolute Molar Flux Mass flux of species B (kg/m2s): For binary mixture of A & B:
24 Example 2Gaseous H2 is stored at elevated pressure in a rectangular container having steel walls 10mm thick. The molar concentration of H2 in the steel at the inner surface is 1 kmol/m3, while the concentration of H2 in the steel at outer surface is negligible. The binary diffusion coefficient for H2 in steel is 0.26x10-12 m2/s. what is the molar and mass diffusive flux for H2 through the steel?
25 Example 2Known: Molar concentration of H2 at inner and outer surfaces of a steel wall. Find: H2 molar and mass flux Schematic:
26 Example 2 Assumptions: Steady-state, 1-D mass transfer conditions CA<<CB (H2 concentration much less than steel), total concentration C=CA+CB is uniformNo chemical reaction between H2 and steelAnalysis:
27 Example 2Analysis:(1). Simplify the molar flux equation
28 Example 2Analysis:(1). Simplify the molar flux equation
34 Evaporation in a Column Assumptions:Ideal gases;No reaction;xA,0>xA,L, xB,0<xB,L;Gas B insoluble in liquid A
35 Evaporation in a Column Stationary or moving medium?
36 Evaporation in a Column Separate variables and integrate
37 Evaporation in a Column Apply B.C.’s to solve C1 & C2:
38 Example 3 An 8-cm-internal-diameter, 30-cm-high pitcher half Filled with water is left in a dry room at 15°C and87 kPa with its top open. If the water is maintainedat 15°C at all times also, determine howlong it will take for the water toevaporate completely.