3_MassDiffusion

3_MassDiffusion - MASS DIFFUSION In this section the mass...

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ME437/537 G. Ahmadi 1 MASS DIFFUSION In this section the mass transfer process is described. The Brownian diffusion of small particles and Fick's law are first discussed. This is followed by the presentation of a number of applications. Brownian Diffusion Small particles suspended in a fluid undergo random translational motions due to molecular collisions. This phenomenon is referred to as the Brownian motion. The Brownian motion leads to diffusion of particles in accordance with Fick’s law. i.e., dx dc D J = ( 1 ) where c is the concentration, J is the flux, and D is the diffusion coefficient. When the effect of particle inertia is negligible, using (1) in the equation of conservation of mass for particles leads to c D c t c 2 = + v ( 2 ) where v is the fluid velocity vector. The particle mass diffusivity is given by d 3 kTC D c πµ = (3) where c C is the Cunningham correction given by (3) and k is the Boltzmann constant (K / erg 10 38 . 1 k 16 × = ). The diffusive may be restated as m kT D τ = ( 4 ) where m is the mass of the spherical particle and τ is its relaxation time. Table 8 – Particle mass diffusivity. ) ( m d µ ) / ( 2 s cm D 10 -2 5.24 × 10 -4 10 -1 6.82 × 10 -6 1 2.74 × 10 -7 10 2.38 × 10 -8

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ME437/537 G. Ahmadi 2 The mean-square displacement for a Brownian particle is given as Dt 2 s 2 = (one-dim) (5) Brownian Motion of Rotation Aerosol particles may also rotate randomly due to the Brownian effects. The mean-square angle of rotation is given as t d kT 2 3 2 πµ = θ (6) Distributions When the gas is in equilibrium, the aerosol particle will have the same average translational energy as molecules. Thus kT 2 3 u m 2 1 2 = , (7) and the root-mean-square particle velocity is given by m / kT 3 u 2 = (8) Under equilibrium, aerosol particles will have a Maxwellian distribution and their concentration in a gravitational field is given by } kT ) x x ( mg exp{ C C 0 0 = (9) Effect of Mass The diffusivity as given by (3) and (4) is independent of particle density, but heavy particles do not respond swiftly to the molecular impacts. A time dependent analysis leads to ] t / ) e 1 ( 1 [ Dt 2 s / t 2 τ τ = (10) where τ is the particle relaxation time. when τ >> t , (10) reduces to Equation (5).
ME437/537 G. Ahmadi 3 Aerosols Mean Free Path The apparent mean free path for aerosol particles α λ is defined as the average distance that a particle moves before changing its direction by 90 o . The average

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This note was uploaded on 01/05/2011 for the course CSU 3 taught by Professor Handsome during the Spring '10 term at CSU Pueblo.

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3_MassDiffusion - MASS DIFFUSION In this section the mass...

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