Molecular Diffusion in Gases

Diffusion plus Convection

)

Molecular Diffusion in Gases

Equimolar Counterdiffusion

A B

BA

𝑐𝐴=𝑐 𝑥𝐴

𝐽 𝐴𝑧∗ =−𝑐𝐷 𝐴𝐵

𝑑𝑥𝐴

𝑑𝑧

In terms of mole fraction,

Molecular Diffusion in Gases

Uni-component Diffusion

http://sst-web.tees.ac.uk/external/U0000504/Notes/ProcessPrinciples/Diffusion/Default.htm

𝑁 𝐴=−𝑐𝐷𝐴𝐵

𝑑𝑥𝐴

𝑑𝑧+𝑐𝐴

𝑁 𝐴

𝑐

𝑁 𝐴=−𝑐𝐷 𝐴𝐵

𝑥𝐵

𝑑𝑥𝐴

𝑑𝑧

Molecular Diffusion in Gases

Example

Water in the bottom of a narrow metal tune is held a t a constant temperature of 293 K. The total pressure of air (assumed dry) is 1.01325 105 Pa and the temperature is 293 K.

Water evaporates and diffuses through the air in the tube, and the diffusion path z2-z1 is 0.1524m long. Calculate the rate of evaporation of water vapor at 293 K and 1 atm pressure. The diffusivity of water in air is 0.250 x 10-4 m2/s. Assume that the system is isothermal.

Introduction to Mass Transfer II

Outline

3. Molecular Diffusion in GasesDiffusion with Varying Cross-sectional Area

4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids6. Prediction of Diffusivities

Molecular Diffusion in Gases

Example: Diffusion through a varying cross-sectional area

A sphere of naphthalene having a radius of 2.0 mm is suspended in a large volume of still air at 318 K and 1.01325x105 Pa. The surface temperature of the naphthalene can be assumed to be at 318 K and its vapor pressure at 318 K is 0.555 mm Hg. The DAB of naphthalene in air at 318 K is 6.92x10-6 m2/s. Calculate the rate of evaporation of naphthalene from the surface.

Molecular Diffusion in Gases

Given:

DAB = 6.92x10-6 m2/spA1 = (0.555/760)*(101325) = 74.0 PapA2 = 0r1 = 0.002 m

* The radius of the sphere decreases slowly with time

𝑁 𝐴=−𝐷 𝐴𝐵𝑃𝑅𝑇 𝑃𝐵

𝑑𝑝𝐴

𝑑𝑧

Molecular Diffusion in Gases

𝑁 𝐴=−𝐷 𝐴𝐵𝑃𝑅𝑇 𝑝𝐵

𝑑𝑝𝐴

𝑑𝑧

ṅ𝐴𝐴

=−𝐷𝐴𝐵𝑃

𝑅𝑇 (𝑃 −𝑝𝐴)𝑑𝑝𝐴

𝑑 𝑟

Where

ṅ𝐴4𝜋 𝑟2

𝑑𝑟=−𝐷 𝐴𝐵𝑃

𝑅𝑇 (𝑃−𝑝𝐴)𝑑𝑝𝐴

Substitution and rearranging,

Molecular Diffusion in Gases

∫𝑟 1

∞ ṅ𝐴4𝜋𝑟 2

𝑑𝑟=−𝐷𝐴𝐵𝑃𝑅𝑇 ∫

𝑝 𝐴1

𝑝𝐴21

(𝑃−𝑝𝐴)𝑑𝑝𝐴

The left side of the equation will be

∫𝑟 1

∞ ṅ𝐴4𝜋𝑟 2

𝑑𝑟=ṅ𝐴4𝜋∫

𝑟 1

∞1𝑟2𝑑𝑟=

ṅ𝐴4 𝜋

[ 1𝑟 1−1∞

]

Molecular Diffusion in Gases

∫𝑟 1

∞ ṅ𝐴4𝜋𝑟 2

𝑑𝑟=−𝐷𝐴𝐵𝑃𝑅𝑇 ∫

𝑝 𝐴1

𝑝𝐴21

(𝑃−𝑝𝐴)𝑑𝑝𝐴

The right side of the equation will be

−𝐷 𝐴𝐵𝑃𝑅𝑇 ∫

𝑝𝐴1

𝑝𝐴21

(𝑃 −𝑝𝐴 )𝑑𝑝𝐴=−

𝐷 𝐴𝐵𝑃𝑅𝑇

𝑙𝑛𝑃 −𝑝𝐴1

𝑃 −𝑝𝐴2

Molecular Diffusion in Gases

∫𝑟 1

∞ ṅ𝐴4𝜋𝑟 2

𝑑𝑟=−𝐷𝐴𝐵𝑃𝑅𝑇 ∫

𝑝 𝐴1

𝑝𝐴21

(𝑃−𝑝𝐴)𝑑𝑝𝐴

Solving for the rate of evaporation,

ṅ𝐴=−4 𝜋𝑟1𝐷 𝐴𝐵𝑃𝑅𝑇

𝑙𝑛𝑃 −𝑝𝐴1

𝑃 −𝑝𝐴2ANS: 4.9 x 10-9 mol/s

Outline

3. Molecular Diffusion in GasesDiffusion with Varying Cross-sectional Area

4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids6. Prediction of Diffusivities

Molecular Diffusion in Liquids

For gases,Kinetic theory is well developed

http://www.bbc.co.uk/bitesize/ks3/science/chemical_material_behaviour/behaviour_of_matter/revision/4/

Gas ModelGases are made of

continuous free space throughout which are

distributed moving molecules.

Molecular Diffusion in Liquids

Liquid Model

A continuous phase of arranged molecules close to

each other but held together by strong intermolecular forces

Dispersed throughout the phase are “holes” of free space

The structure is more complex.

Molecular Diffusion in Liquids

Rate of DiffusionBUT only about 100 times faster….

Molecular Diffusion in Liquids

Equations for Diffusion

)

1. For equimolarcounterdiffusion,

𝑁 𝐴=−𝑐𝐷𝐴𝐵

𝑑𝑥𝐴

𝑑𝑧

where

Molecular Diffusion in Liquids

Equations for Diffusion

)

2. For unicomponent diffusion,

𝑁 𝐴=−𝐷𝐴𝐵 (1+𝑐𝐴

𝑐𝐵

)𝑑𝑐 𝐴

𝑑𝑧 𝑐𝐴+𝑐𝐵=¿NOTE:

average value for the molar density of the mixture

Molecular Diffusion in Liquids

Example

An ethanol (A) – water (B) solution in the form of a stagnant film 2.0 mm thick at 293 K is in contact at one surface with an organic solvent in which ethanol is soluble and water is insoluble. Hence NB = 0. At point 1 the concentration of ethanol is 16.8 wt% and the solution density ρ1 = 972.8 kg/m3. At point 2 ethanol concentration is 6.8 wt% and ρ2 = 988.1 kg/m3. The diffusivity of ethanol is 0.740x10-9 m2/s. Calculate the steady-state flux NA.

Outline

3. Molecular Diffusion in GasesDiffusion with Varying Cross-sectional Area

4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids6. Prediction of Diffusivities

Molecular Diffusion in Solids

Rate of DiffusionWhat do we expect?

Outline

3. Molecular Diffusion in GasesDiffusion with Varying Cross-sectional Area

4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids6. Prediction of Diffusivities

Predicting Diffusivities

For gases at low density- almost independent of concentration- increase with temperature - vary inversely with pressure

For liquids and solids, - strongly concentration-dependent - generally increase with temperature

Predicting Diffusivities

Empirical EquationsFor gases,1. See Table 2-324 Perry’s2. Chapman and Enskog Equation

DAB = diffusivity in m2/sT = temperature in KMA = molecular weight of A in kg/kmol

MB = molecular weight of B in kg/kmolσAB = average collision diameterΩD,AB= collision integral based on Lennard-Jones potential

Predicting Diffusivities

Empirical EquationsFor gases,3. Gilliland Equation

DAB = diffusivityT = temperature

MA = molecular weightV = molar volume

P= pressure

𝐷 𝐴𝐵=1.38 𝑥10− 7√𝑇 3( 1

𝑀𝑎

+ 1𝑀𝑏

)

𝑃 (𝑉 𝑎

13+𝑉 𝑏

13 )2

Predicting Diffusivities

Empirical EquationsFor liquids,4. See Table 2-325 Perry’s5. Stokes Einstein Equation

4. Wilke and Chang Equation

Predicting Diffusivities

All diffusivities have units m2/sTherefore, their ratios are dimensionless groups

Dim. Group Ratio Equation

Prandtl, Pr molecular diffusivity of momentum / molecular diffusivity of heat

Schmidt, Sc momentum diffusivity/ mass diffusivity

Lewis, Le thermal diffusivity/ mass diffusivity