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Diffusion Mass Transfer Chapter 14 Sections 14.1 through 14.7 Lecture 18

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1.Physical Origins and Rate Equations 2.Mass Transfer in Nonstationary Media 3.Conservation Equation and Diffusion through Stationary Media 4.Diffusion and Concentrations at Interfaces 5.Diffusion with Homogenous Reactions 6.Transient Diffusion

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1. Physical Origins and Rate Equations 1.Driving potential for mass transfer Concentration Gradient 2.Modes of mass transfer Convection and Diffusion

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General Considerations 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 decreasing concentration dictates net transport of A molecules to the right and B molecules to the left. – In time, uniform concentrations of A and B are achieved. Mass transfer refers to mass in transit due to a species concentration gradient in a mixture.

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Definitions Molar concentration of species i. Mass density (kg/m 3 ) 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. Transport of i relative to a fixed reference frame. Mole fraction of species i Mass fraction of species i Absolute mass flux of species i.

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Mixture Composition Definitions: Mass density of species i: i =M i *C i (kg/m 3 ) Molecular weight of species i: M i (kg/kmol) Molar concentration of species i: C i (kmol/m 3 ) Mixture mass density: (kg/m 3 ) Total number of moles per unit volume of mixture

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Mixture Composition Definitions: Mass fraction of species i: m i = i / Molar fraction of species i: x i =C i /C For ideal gases:

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Fick’s Law of Diffusion For transfer of species A in a binary mixture of A & B Mass flux of species A (kg/m 2 s): Molar flux of species A (mol/m 2 s): Binary diffusivity D AB (m 2 /s) Ordinary diffusion due to concentration gradient and relative to coordinates that move with average velocity

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Fick’s Law of Diffusion For transfer of species A in a binary mixture of A & B Mass flux of species A (kg/m 2 s): Molar flux of species A (mol/m 2 s): If C and are constants, the above equations become:

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Mass Diffusivity For ideal gases Gas-GasLiquid- Liquid Gas in Solid ~10 -5 m 2 /s~10 -9 m 2 /s~ m 2 /s Solid in Solid ~ m 2 /s At 298 K, 1 atm

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Example 14.1 Consider 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 dC A /dx= 1 kmol/m 3 ● m. Compare the value of the mass diffusivity to the thermal diffusivity. The mole fraction of the hydrogen x A, is much less than unity.

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Example 14.1 Known: Concentration gradient of hydrogen in air, liquid water, or iron at 293 K Find: Molar and mass fluxes of hydrogen and the relative values of the mass diffusivity and thermal diffusivity Schematic:

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Example 14.1 Assumptions: Steady-state conditions Properties: Table A.8, hydrogen-air (298 K): D AB = 0.41x10 -4 m 2 /s, hydrogen- water (298 K): D AB = 0.63x10 -8 m 2 /s, hydrogen-iron (293 K): D AB = 0.26x m 2 /s. Table A.4, air (293 K): α= 21.6x10 -6 m 2 /s; Table A.6, water (293 K): k = W/mK, ρ = 998 kg/m 3, c p = 4182 J/kg K. Table A.l, iron (300 K): α = 23.1 x m 2 /s. Analysis: Using Eqn , we can find that the mass diffusivity of hydrogen in air at T=293K is

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Example 14.1 For the case where hydrogen is a dilute species, that is x A <<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 The molar flux of hydrogen is described by Fick’s law, Equation

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Example 14.1 Hence, for the hydrogen-air mixture, The mass flux of hydrogen in air is found to from the expression:

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Example 14.1 The results for the three different mixtures are summarized in the following table:

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Mass Transfer in Nonstationary Media For mass flux relative to a fixed coordinate system Mass flux of species A (kg/m 2 s): Mass flux of species B (kg/m 2 s): Mass flux of mixture (kg/m 2 s): Mass-average velocity (m/s): Absolute Mass Flux

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Relative Mass Flux Mass flux of species A (kg/m 2 s):

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Relative Mass Flux Mass flux of species A (kg/m 2 s): For binary mixture of A & B:

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Relative Mass Flux For binary mixture of A & B: D AB =D BA

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Absolute Molar Flux For molar flux relative to a fixed coordinate system Molar flux of species A (mol/m 2 s): Mass flux of species B (kg/m 2 s): Mass flux of mixture (kg/m 2 s): Molar-average velocity (m/s):

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Absolute Molar Flux Molar flux of species A (kg/m 2 s): Mass flux of species A (kg/m 2 s):

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Absolute Molar Flux Mass flux of species B (kg/m 2 s): For binary mixture of A & B:

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Example 2 Gaseous H 2 is stored at elevated pressure in a rectangular container having steel walls 10mm thick. The molar concentration of H 2 in the steel at the inner surface is 1 kmol/m 3, while the concentration of H 2 in the steel at outer surface is negligible. The binary diffusion coefficient for H 2 in steel is 0.26x m 2 /s. what is the molar and mass diffusive flux for H 2 through the steel?

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Example 2 Known: Molar concentration of H 2 at inner and outer surfaces of a steel wall. Find: H 2 molar and mass flux Schematic:

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Example 2 Assumptions: Steady-state, 1-D mass transfer conditions C A <

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Example 2 Analysis: (1). Simplify the molar flux equation

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Example 2 Analysis: (1). Simplify the molar flux equation 0

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Example 2 Analysis: (2). Apply C = constant

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Example 2 Analysis: (3). From mass conservation:

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Example 2 Analysis: (4). Integration from x=0 to x=L, C A =C A,1 to C A,2

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Example 2 Analysis: (5). Mass flux

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Evaporation in a Column

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Assumptions: 1.Ideal gases; 2.No reaction; 3.x A,0 >x A,L, x B,0

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Evaporation in a Column Stationary or moving medium?

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Evaporation in a Column Separate variables and integrate

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Evaporation in a Column Apply B.C.’s to solve C 1 & C 2 :

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Example 3 An 8-cm-internal-diameter, 30-cm-high pitcher half Filled with water is left in a dry room at 15°C and 87 kPa with its top open. If the water is maintained at 15°C at all times also, determine how long it will take for the water to evaporate completely.

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Example 3

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Lecture 18

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