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Molecular Diffusion in Gases Diffusion plus Convection ππ΄ = ππ₯π΄ −ππ·π΄π΅ ππ§ + ππ΄ +ππ΅ ππ΄ ( ) π Molecular Diffusion in Gases Equimolar Counterdiffusion In terms of mole fraction, A B ππ΄ = ππ₯π΄ ∗ π½π΄π§ A B ππ₯π΄ = −ππ·π΄π΅ ππ§ Molecular Diffusion in Gases Uni-component Diffusion ππ₯π΄ ππ΄ ππ΄ = −ππ·π΄π΅ + ππ΄ ππ§ π ππ·π΄π΅ ππ₯π΄ ππ΄ = − π₯π΅ ππ§ http://sst-web.tees.ac.uk/external/U0000504/Notes/ProcessPrinciples/Diffusion/Default.htm 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 ο΄ 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 Gases Diffusion with Varying Cross-sectional Area 4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids 6. 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/s pA1 = (0.555/760)*(101325) = 74.0 Pa pA2 = 0 r1 = 0.002 m π·π΄π΅ π πππ΄ ππ΄ = − π πππ΅ ππ§ * The radius of the sphere decreases slowly with time Molecular Diffusion in Gases π·π΄π΅ π πππ΄ ππ΄ = − π πππ΅ ππ§ αΉ π΄ π·π΄π΅ π πππ΄ =− π΄ π π(π − ππ΄ ) ππ Where A = 4ππ 2 Substitution and rearranging, αΉ π΄ π·π΄π΅ π ππ = − πππ΄ 2 4ππ π π(π − ππ΄ ) Molecular Diffusion in Gases ∞ π1 αΉ π΄ π·π΄π΅ π ππ = − 2 4ππ π π ππ΄2 ππ΄1 1 πππ΄ (π − ππ΄ ) The left side of the equation will be ∞ π1 αΉ π΄ αΉ π΄ ππ = 2 4ππ 4π ∞ π1 1 αΉ π΄ 1 1 ππ = [ − ] 2 π 4π π1 ∞ Molecular Diffusion in Gases ∞ π1 αΉ π΄ π·π΄π΅ π ππ = − 2 4ππ π π ππ΄2 ππ΄1 1 πππ΄ (π − ππ΄ ) The right side of the equation will be π·π΄π΅ π − π π ππ΄2 ππ΄1 1 π·π΄π΅ π π − ππ΄1 πππ΄ = − ππ π − ππ΄ π π π − ππ΄2 Molecular Diffusion in Gases ∞ π1 αΉ π΄ π·π΄π΅ π ππ = − 2 4ππ π π ππ΄2 ππ΄1 1 πππ΄ (π − ππ΄ ) Solving for the rate of evaporation, π·π΄π΅ π π − ππ΄1 αΉ π΄ = −4ππ1 ππ π π π − ππ΄2 ANS: 4.9 x 10-9 mol/s Outline 3. Molecular Diffusion in Gases Diffusion with Varying Cross-sectional Area 4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids 6. Prediction of Diffusivities Molecular Diffusion in Liquids Gas Model Gases are made of continuous free space throughout which are distributed moving molecules. For gases, Kinetic theory is well developed ππ ππππ β« ππππ http://www.bbc.co.uk/bitesize/ks3/science/chemical_material_behaviour/behaviour_of_matter/revision/4/ 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 Diffusion πΊππ ππ > πΏπππ’πππ BUT only about 100 times faster…. Molecular Diffusion in Liquids Equations for Diffusion ππ΄ = ππ₯π΄ −ππ·π΄π΅ ππ§ + ππ΄ +ππ΅ ππ΄ ( ) π 1. For equimolarcounterdiffusion, ππ΄ = −ππ΅ ππ₯π΄ ππ΄ = −ππ·π΄π΅ ππ§ where π π π π = = ( )π΄ + ( )π΅ π π π Molecular Diffusion in Liquids Equations for Diffusion ππ΄ = ππ₯π΄ −ππ·π΄π΅ ππ§ + ππ΄ +ππ΅ ππ΄ ( ) π 2. For unicomponent diffusion, ππ΅ = 0 ππ΄ πππ΄ ππ΄ = −π·π΄π΅ (1 + ) ππ΅ ππ§ NOTE: π π ππ΄ + ππ΅ = ( )π π ( )π ο average value for the π 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 Gases Diffusion with Varying Cross-sectional Area 4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids 6. Prediction of Diffusivities Molecular Diffusion in Solids What do we expect? Rate of Diffusion πΊππ ππ > πΏπππ’πππ > ππππππ Outline 3. Molecular Diffusion in Gases Diffusion with Varying Cross-sectional Area 4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids 6. 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 Equations For gases, 1. See Table 2-324 Perry’s 2. Chapman and Enskog Equation DAB = diffusivity in m2/s T = temperature in K MA = molecular weight of A in kg/kmol MB = molecular weight of B in kg/kmol σAB = average collision diameter β¦D,AB= collision integral based on Lennard-Jones potential Predicting Diffusivities Empirical Equations For gases, 3. Gilliland Equation π·π΄π΅ = 1 1 1.38 π₯ 10−7 π 3 (π + π ) π π 1 π(ππ3 + 1 ππ3 )2 DAB = diffusivity T = temperature MA = molecular weight V = molar volume P= pressure Predicting Diffusivities Empirical Equations For liquids, 4. See Table 2-325 Perry’s 5. Stokes Einstein Equation 4. Wilke and Chang Equation Predicting Diffusivities All diffusivities have units m2/s Therefore, 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 ππ π π ν π·π΄π΅ πΌ π·π΄π΅
