! Introduction to Aerosols Introduction to Aerosols ! ! Drag Forces Drag Forces ! ! Cunningham Corrections Cunningham Corrections ! ! Lift Forces Lift Forces ! ! Brownian Motion Brownian Motion ! ! Particle Deposition Mechanisms Particle Deposition Mechanisms ! ! Gravitational Sedimentation Gravitational Sedimentation ! ! Aerosol Coagulation Aerosol Coagulation ! ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi � Equivalent area diameters � Equivalent area diameters � Aerosols are suspension of � Aerosols are suspension of solid or solid or � Feret Feret’ ’s s diameter diameter (maximum distance � liquid particles in a gas. (maximum distance liquid particles in a gas. � edge to edge) � edge to edge) � Dust, smoke, mists, fog, haze, and � Dust, smoke, mists, fog, haze, and � Stokes � Stokes’ ’ diameter diameter (diameter of a (diameter of a smog are common aerosols. . smog are common aerosols sphere with the same density and the sphere with the same density and the � Aerosol particles are found � Aerosol particles are found same velocity as the particle) same velocity as the particle) in different shapes. . in different shapes � Aerodynamic diameter � Aerodynamic diameter (diameter of (diameter of a sphere with the density of water a sphere with the density of water and the same velocity as the particle) and the same velocity as the particle) ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 1
Aerosols Air Knudsen Number λ Knudsen Number 2 = Kn d Number Density 100-10 5 10 19 (Number/cm) Mach Number Mach Number v − p f | v | = M f c Mean Temperature 240 – 310 240 – 310 (K) Schmidt Number Schmidt Number ν f λ 2 n d = = Sc Mean Free Path Greater than 1 m 0.06 µ m D 4 2 � 10 -4 µ m Brown Number 0.01 – 10 µ m Brown Number Particle Radius v p , 2 | v ' p | = 1 / 2 = Br ( ) f , 2 f v | v ' | 10 -18 - 10 -9 Particle Mass (g) 4.6 � 10 -23 p − f Reynolds Number Reynolds Number | v v | d 4 M = = Re ν K Particle Charge 0 – 100 Weakly Ionized n (Elementary Charge Single Charge Units) ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi λ = Mean Free Path λ = Mean Free Path ν = = Kinematic Kinematic Viscosity Viscosity ν 1 kT λ = = d = Particle Diameter d = Particle Diameter D = Diffusivity D = Diffusivity π 2 π 2 2 nd 2 d P m m v = Particle Velocity v’ ’ = Thermal Velocity = Thermal Velocity v = Particle Velocity v p p v = Fluid (Air) Velocity n = Number Density Molecular Molecular v = Fluid (Air) Velocity n = Number Density f f = × -23 k 1.38 10 J / K f f Diameter Diameter c = Speed of Sound c = Speed of Sound 23 . 1 T λ µ = Air ( m ) Air P ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 2
µm Particle Diameter, µm Particle Diameter, 4 10 − 4 10 − 3 10 − 2 10 − 1 0 1 2 3 10 10 10 10 10 4 10 − 10 − 10 − 10 − 10 4 3 2 1 10 0 10 1 10 2 10 3 Electro. Infrared Microwaves X-Ray UV Vis Ultrasonic Settling Chamber Wave Centrifuge Gas Solid Fume Dust Definition Mist Spray Cleaning Air Filter Liquid Method HE Air Filter Impact Separator Clay Silt Sand Gravel Soil Thermal Separator Atmospheric Smog Cloud/Fog Mist Rain Electrostatic Separator Typical Viruses Bacteria Hair × − 2 10 − 5 × − 9 × − 11 5 10 2 10 2 10 Air Diffusion Particles Smoke Coal Dust Beach Sand Coeff. cm 2 /s Water × − 12 × − × − × − 5 10 5 10 6 5 10 8 5 10 10 Microscopy Size × − 10 − 6 2 10 4 0 . 6 Electron Microscopy Sieving Terminal 600 Analysis Air Velocity cm/s Ultra Centrifuge Sedimentation methods − 10 − 10 × − 7 × 3 Water 6 10 6 10 12 S=2 ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi π F = 3 µUd Stokes Stokes C D F 24 = = C D Drag Drag D 1 Re ρ 2 Coefficient Coefficient U A 2 Reynolds Reynolds ρ = Ud Re Number Number Re µ ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 3
+ Laminar Laminar 24 [ 1 3 Re/ 16 ] = C D Oseen Oseen Boundary Layer Boundary Layer Re + 0 . 687 24 [ 1 0 . 15 Re ] < < 1 Re 1000 = C D Re Turbulent Turbulent Newton Newton Boundary Layer Boundary Layer C D = 0 . 4 3 < < × 5 10 Re 2 . 5 10 ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 1000 Normal to the Wall Normal to the Wall d 100 ( (Bernner Bernner, 1961) , 1961) Experiment h U CD Oseen 10 Eq. (5) Stokes Newton 1 0 24 d 0 1 10 100 1000 10000 = + C D ( 1 ) Re Re 2 h Predictions of various models for drag coefficient for a spherical particle. Predictions of various models for drag coefficient for a spheric al particle. ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 4
U For 1000 > For 1000 > Kn Kn > 0 > 0 Normal to the Wall Normal to the Wall d πµ 3 Ud (Faxon ( Faxon, 1923) , 1923) Stokes Stokes- -Cunningham Cunningham = F D Drag Drag h C c Cunningham Cunningham Correction Correction λ 24 9 d 1 d 45 d 1 d 2 = − + − − − C [ 1 ( ) ( ) 3 ( ) 4 ( ) 5 ] 1 = + + − λ 1 . 1 d / 2 C 1 [ 1 . 257 0 . 4 e ] D Re 16 2 h 8 2 h 256 2 h 16 2 h c d ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi c Variations of C c with d for λ λ = 0.07 = 0.07 µ µm m 1000 Variations of C c with d for Diameter, µm C 100 c 10 µm 1.018 Cc 10 1 µm 1.176 0.1 µm 3.015 1 0.01 µm 23.775 0.001 0.01 0.1 1 10 100 Kn 0.001 µm 232.54 Variation of Cunningham correction with Knudsen number. Variation of Cunningham correction with Knudsen number. ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 5
Henderson (1976) Henderson (1976) M < 1 M < 1 Henderson (1976) Henderson (1976) M > 1 M > 1 − 1 ⎡ ⎤ ⎧ ⎫ ⎛− Re ⎞ = + + × C 24 Re S ⎨ 4 . 33 1 . 567 exp ⎜ 0 . 247 ⎟ ⎬ ⎢ ⎥ 1 ⎡ ⎤ D ⎛ ⎞ ⎩ ⎝ S ⎠ ⎭ 0 . 34 M 2 2 1 . 058 1 ⎣ ⎦ + + + + − 0 . 9 1 . 86 ⎜ ⎟ ⎢ 2 ⎥ ( ) ⎡ ⎤ 2 2 4 + + M ⎝ Re ⎠ S S S ⎛− 0 . 5 M ⎞ 4 . 5 0 . 38 0 . 03 Re 0 . 48 Re ⎢ ⎥ ⎣ ⎦ + + 2 + 8 + exp ⎜ ⎟ 0 . 1 M 0 . 2 M ⎢ ⎥ = C + + ⎝ Re ⎠ 1 0 . 03 Re 0 . 48 Re ⎣ ⎦ D 1 ⎛ ⎞ M 2 ⎜ ⎟ ⎡ ⎤ + ⎛− M ⎞ 1 1 . 86 ⎜ ⎟ + − 1 exp ⎜ ⎟ 0 . 6 S ⎢ ⎥ Re ⎝ ⎠ ⎝ Re ⎠ ⎣ ⎦ ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi Carlson and Hoglund Carlson and Hoglund (1964) (1964) + µ µ f p 1 2 / 3 = πµ f F 3 Ud D + µ µ f p 1 / 0 . 427 3 + − − 1 exp( } 24 4 . 63 0 . 88 M Re = C D M Re Re + + − 1 { 3 . 82 1 . 28 exp( 1 . 25 )} = πµ f Re M F 2 Ud For Bubbles For Bubbles D ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 6
Cluster Correction Cluster Correction Cluster Correction Shape Shape Shape = πµ F 3 Ud K K = 1.12 K = 1.32 K = 1.17 oo oooo oo D e oo ooo K = 1.27 ooooo K = 1.45 o o K = 1.19 o 6 = 1 / 3 d ( Volume ) o o e π K = 1.16 K = 1.57 oo K = 1.17 o oooooo oo o o oo oooooo K = 1.64 ooooooo K = 1.73 K=Correction Factor K=Correction Factor o o ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi a b π β = F D = 6 µUaK' β = b a b b b a a 4 β − ( 2 1 ) a 3 = K' β − ( 2 2 1 ) ln[ β + ( β 2 − 1 ) 1 / 2 ] − β β 2 − 1 / 2 ( 1 ) b 8 β 2 − 4 ( 1 ) 8 β − ( 2 1 ) β − ( 2 1 ) 3 K' = 3 a 3 = K' = K' β 2 − ( 2 3 ) β β 2 − ( 2 ) β ( 3 β 2 − 2 ) β + β 2 − 1 / 2 + β ln[ ( 1 ) ] − tan 1 ( β 2 − 1 ) 1 / 2 ] + β − 1 β 2 − 1 / 2 − β tan ( 1 ) ] β − ( 2 1 ) 1 / 2 β − ( 2 1 ) 1 / 2 β 2 − 1 / 2 ( 1 ) ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 7
πµ 4 Ub = F D = µ b F D 16 aU a β ln 2 b πµ 8 Ub = F D = µ F D 32 aU / 3 a β ln 2 ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi Drag Gravity Equation of Motion Equation of Motion πµ 4 U πµ p = d u 3 d F 2 aU = f − p + D = − R e m ( u u ) m g ( 2 . 002 ln R ) ν e dt C c ME 437/537-Particle G. Ahmadi ME 437/537-Particle G. Ahmadi 8
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