! Introduction to Aerosols Introduction to Aerosols ! ! Drag Forces - - PowerPoint PPT Presentation

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! Introduction to Aerosols Introduction to Aerosols ! ! Drag Forces - - PowerPoint PPT Presentation

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! 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

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SLIDE 1

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! ! 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

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  • Aerosols are suspension of

Aerosols are suspension of solid or solid or liquid particles in a gas. liquid particles in a gas.

  • Dust, smoke, mists, fog, haze, and

Dust, smoke, mists, fog, haze, and smog are common aerosols smog are common aerosols. .

  • Aerosol particles are found

Aerosol particles are found in different shapes in different shapes. .

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  • Equivalent area diameters

Equivalent area diameters

  • Feret

Feret’ ’s s diameter diameter (maximum distance

(maximum distance

  • edge to edge)

edge to edge)

  • Stokes

Stokes’ ’ diameter diameter (diameter of a

(diameter of a sphere with the same density and the sphere with the same density and the same velocity as the particle) same velocity as the particle)

  • 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)

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SLIDE 2

2

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Aerosols Air

Number Density (Number/cm) 100-105 1019 Mean Temperature (K) 240 – 310 240 – 310 Mean Free Path Greater than 1 m 0.06 µm Particle Radius 0.01 – 10 µm 2 10-4 µm Particle Mass (g) 10-18 - 10 -9 4.6 10-23 Particle Charge (Elementary Charge Units) 0 – 100 Weakly Ionized Single Charge

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d 2 Kn λ =

f

c | | M

f p

v v − =

4 d n D Sc

2 fλ

= ν = | ' v | | ' v | ) v v ( Br

f p 2 / 1 2 , f 2 , p

= =

n

K M 4 d | | Re = ν − =

f p

v v

Knudsen Number Knudsen Number Mach Number Mach Number Schmidt Number Schmidt Number Brown Number Brown Number Reynolds Number Reynolds Number

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p f f

λ λ = Mean Free Path = Mean Free Path ν ν = = Kinematic Kinematic Viscosity Viscosity d = Particle Diameter d = Particle Diameter D = Diffusivity D = Diffusivity v = Particle Velocity v = Particle Velocity v v’ ’ = Thermal Velocity = Thermal Velocity v = Fluid (Air) Velocity v = Fluid (Air) Velocity n = Number Density n = Number Density c = Speed of Sound c = Speed of Sound

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p f f

P d 2 kT nd 2 1

2 m 2 m

π = π = λ

P T 1 . 23 ) m ( = µ λ

K / J 10 1.38 k

  • 23

× =

Molecular Molecular Diameter Diameter Air Air

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SLIDE 3

3

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4

10

3

10

2

10

1

10 10

1

10−

2

10−

3

10−

Particle Diameter,

4

10− Electro. Wave

X-Ray µm UV Vis Infrared Microwaves Fume

Definition

Dust Mist Spray Solid Liquid

Soil Atmospheric Typical Particles Size Analysis methods

Clay Silt Sand Gravel Smog Cloud/Fog Mist Rain Viruses Bacteria Hair Smoke Coal Dust Beach Sand Microscopy Electron Microscopy Sieving Ultra Centrifuge Sedimentation

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4

10

3

10

2

10

1

10 10

1

10−

2

10−

3

10−

Particle Diameter,

4

10− Gas Cleaning Method

µm Ultrasonic Settling Chamber Centrifuge Air Filter

Diffusion

  • Coeff. cm2/s

Impact Separator Electrostatic Separator HE Air Filter Thermal Separator

Terminal Velocity cm/s S=2 Air Water Air Water

2

10 5

−

×

5

10−

9

10 2

−

×

11

10 2

−

×

12

10 5

−

×

10

10 5

−

×

8

10 5

−

×

6

10 5

−

×

6

10−

4

10 2

−

×

7

10 6

−

×

3

10 6

−

×

10

10− 6 . 600 12

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µUd 3 = F π

Stokes Stokes

Re 24 A U 2 1 F C

2 D D

= ρ =

Drag Drag Coefficient Coefficient

µ ρ = Ud Re

Reynolds Reynolds Number Number

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ME 437/537-Particle

D

C

Re

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SLIDE 4

4

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Oseen Oseen

Re ] 16 Re/ 3 1 [ 24 CD + =

Re ] Re 15 . 1 [ 24 C

687 . D

+ =

4 . CD =

1000 Re 1 < <

5 3

10 5 . 2 Re 10 × < <

Newton Newton

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Turbulent Turbulent Boundary Layer Boundary Layer Laminar Laminar Boundary Layer Boundary Layer

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1 10 100 1000

CD

1 10 100 1000 10000

Re

Stokes

  • Eq. (5)

Oseen Newton Experiment

Predictions of various models for drag coefficient for a spheric Predictions of various models for drag coefficient for a spherical particle. al particle.

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U d h

Normal to the Wall Normal to the Wall ( (Bernner Bernner, 1961) , 1961)

) h 2 d 1 ( Re 24 CD + =

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SLIDE 5

5

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U

d h

1 5 4 3 D

] ) h 2 d ( 16 1 ) h 2 d ( 256 45 ) h 2 d ( 8 1 ) h 2 d ( 16 9 1 [ Re 24 C

−

− − + − = Normal to the Wall Normal to the Wall ( (Faxon Faxon, 1923) , 1923)

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For 1000 > For 1000 > Kn Kn > 0 > 0

c D

C Ud 3 F πµ = ] e 4 . 257 . 1 [ d 2 1 C

2 / d 1 . 1 c λ −

+ λ + =

Stokes Stokes-

  • Cunningham

Cunningham Drag Drag Cunningham Cunningham Correction Correction

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1 10 100 1000

Cc

0.001 0.01 0.1 1 10 100

Kn Variation of Cunningham correction with Knudsen number. Variation of Cunningham correction with Knudsen number.

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c c

Diameter, µm C 10 µm 1.018 1 µm 1.176 0.1 µm 3.015 0.01 µm 23.775 0.001 µm 232.54

Variations of C Variations of Cc

c with d for

with d for λ λ = 0.07 = 0.07 µ µm m

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SLIDE 6

6

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( )

S 6 . Re M exp 1 M 2 . M 1 . Re 48 . Re 03 . 1 Re 48 . Re 03 . 38 . 5 . 4 Re M 5 . exp S Re 247 . exp 567 . 1 33 . 4 S Re 24 C

8 2 1 D

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− − + + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + + + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− × + + =

−

Henderson (1976) Henderson (1976) M < 1 M < 1

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Henderson (1976) Henderson (1976) M > 1 M > 1

2 1 4 2 2 1 2 D

Re M 86 . 1 1 S 1 S 058 . 1 S 2 2 Re M 86 . 1 M 34 . 9 . C ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + =

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Carlson and Carlson and Hoglund Hoglund (1964) (1964)

)} M Re 25 . 1 exp( 28 . 1 82 . 3 { Re M 1 } Re 3 M 427 . exp( 1 Re 24 C

88 . 63 . 4 D

− + + − − + =

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p f p f f D

/ 1 3 / 2 1 Ud 3 F µ µ + µ µ + πµ = Ud 2 F

f D

πµ =

For Bubbles For Bubbles

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SLIDE 7

7

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K=Correction Factor K=Correction Factor

K Ud 3 F

e D

πµ =

3 / 1 e

) Volume 6 ( d π =

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ME 437/537-Particle Cluster Shape Correction Cluster Shape Correction Cluster Shape Correction

  • K = 1.12
  • K = 1.32
  • K = 1.17
  • K = 1.27
  • K = 1.45
  • o
  • o

K = 1.19

  • o

K = 1.16

  • K = 1.57
  • K = 1.17
  • K = 1.64
  • K = 1.73
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µUaK' 6 = FD π

b a b a β − − β + β − β − β − β = ] ) 1 ( ln[ ) 1 ( ) 1 2 ( ) 1 ( 3 4 K'

2 / 1 2 2 / 1 2 2 2

β + − β + β − β − β − β = ] ) 1 ( ln[ ) 1 ( ) 3 2 ( ) 1 ( 3 8 K'

2 / 1 2 2 / 1 2 2 2

a b = β

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ME 437/537-Particle

a b a b

β + − β − β − β β − β =

−

] ) 1 ( tan ) 1 ( ) 2 ( ) 1 ( 3 4 K'

2 / 1 2 1 2 / 1 2 2 2

β − − β − β − β β − β =

−

] ) 1 ( tan ) 1 ( ) 2 3 ( ) 1 ( 3 8 K'

2 / 1 2 1 2 / 1 2 2 2

b a = β

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SLIDE 8

8

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a a

aU 16 FD µ =

3 / aU 32 FD µ =

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ME 437/537-Particle

b b

β πµ = 2 ln Ub 4 FD β πµ = 2 ln Ub 8 FD

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ME 437/537-Particle

) R ln 002 . 2 ( U 4 F

e D

− πµ =

ν = aU 2 R e

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ME 437/537-Particle

Drag Gravity

g u u u

p f p

m ) ( C d 3 dt d m

c

+ − πµ =

Equation of Motion Equation of Motion

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SLIDE 9

9

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g u u u

p f p

τ + − = τ ) ( dt d

ν = µ ρ = πµ = τ 18 C Sd 18 C d d 3 mC

c 2 c p 2 c

Relaxation Time Relaxation Time

f p

S ρ ρ =

) m ( d 10 3 ) s (

2 6

µ × ≈ τ

−

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ME 437/537-Particle

) e 1 )( (

/ t f p τ −

− τ + = g u u

µ ρ = τ = 18 gC d g u

c 2 p t

Terminal Velocity = Equilibrium Velocity after Large Time Terminal Velocity = Equilibrium Velocity after Large Time

) m ( d 30 ) s / m ( u

2 t

µ ≈ µ

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Stopping Distance = Penetration distance for Stopping Distance = Penetration distance for an initial velocity of an initial velocity of u uo

  • )

e 1 (

/ t τ −

− τ =

p

  • p

u x

τ −

=

/ t

e

  • p

u u

τ =

p

  • p

u x

) m ( d 3 ) m ( x

2 p

µ ≈ µ

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ME 437/537-Particle

76.2 mm 7.62 mm 7.6×10-3 7.47 cm/s 50 3.09 mm 309 µm 3.1×10-4 3.03 mm/s 10 0.786 mm 78.6 µm 7.9×10-5 0.77 mm/s 5 0.0357 mm 3.6 µm 3.6×10-6 35 µm/s 1 0.0103 mm 1.03 µm 1×10-6 10.1 µm/s 0.5 0.0009 mm 0.092 µm 9.1×10-8 0.93 µm/s 0.1 0.0004 mm 0.04 µm 4×10-8 0.39 µm/s 0.05 Stopping Distance u= 10 m/s Stopping Distance u= 1 m/s τ sec Terminal Velocity Diameter, µm

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SLIDE 10

10

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ME 437/537-Particle

)] e 1 ( t )[ ( ) e 1 (

/ t f / t τ − τ −

− τ − τ + + − τ + = g u u x x

p

  • p
  • p

)] e 1 ( / t [ u / x

/ t f p τ −

− − τ = τ

)] e 1 ( / t [ u / y

/ t f p τ −

− − τ α − = τ

τ τ = α

f

u g Components Components

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ME 437/537-Particle

  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

y/utau

1 2 3 4 5 6

t/tau

α =0.1 α =1 α =2

Variations of the particle vertical position with time. Variations of the particle vertical position with time.

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ME 437/537-Particle

  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

y/utau

1 2 3 4 5 6

x/utau

α =0.1 α =1 α =2

Sample particle trajectories. Sample particle trajectories.

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ME 437/537-Particle

g u u u

p f p

) m m ( ) ( C d 3 dt d ) m m (

f c a

− + − πµ = + 6 d m

f 3 f

ρ π =

. m 2 1 m

f a =

Fluid Mass Fluid Mass Apparent Mass Apparent Mass

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SLIDE 11

11

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ME 437/537-Particle

) S 1 1 ( ) ( dt d ) S 2 1 1 ( − τ + − = τ + g u u u

p f p

) 1 ( 18 gC d ) S 1 1 ( g u

p f c 2 p t

ρ ρ − µ ρ = − τ =

Terminal Velocity Terminal Velocity

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ME 437/537-Particle

Lift uf up

) dy du sgn( | dy du | ) u u ( d 615 . 1 F

f 2 / 1 f p f 2 2 / 1 ) Saff ( L

− ρν =

Saffman Saffman (1965, 1968) (1965, 1968)

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ME 437/537-Particle

1 d | u u | R

p f es

<< ν − = 1 d R

2 eG

<< ν γ = &

1 d R

2 e

<< ν Ω =

Ω

1 R R ε

es 2 / 1 eG >>

=

McLaughlin (1991) McLaughlin (1991)

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ME 437/537-Particle

⎩ ⎨ ⎧ > α ≤ α + − α − = 40 R for ) R ( 0524 . 40 R for 3314 . ) 10 / R exp( ) 3314 . 1 ( F F

es 2 / 1 es es 2 / 1 es 2 / 1 ) Saff ( L L es eG 2 es p f

R 2 R 2 R | u u | 2 d = ε = − γ = α &

)]} 32 . ( 6 tan[ 667 . )]}{ 191 . ( log 5 . 2 tanh[ 1 { 3 . F F

10 ) Saff ( L L

− ε + + ε + =

20 0.1 ≤ ε ≤

Mai (1992) Mai (1992)

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SLIDE 12

12

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⎩ ⎨ ⎧ << ε ε ε − >> ε ε − =

− −

1 for ) ln( 140 1 for 287 . 1 F F

2 5 2 ) Saff ( L L

McLaughlin (1991) McLaughlin (1991)

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ME 437/537-Particle

Cherukat Cherukat and McLaughlin (1994) and McLaughlin (1994)

4 / I d V F

L 2 2

) L C ( L

ρ =

−

l u u u V

p f p

γ − = − = &

V 2 d

G

γ = ∧ & l 2 d K =

) K , ( I I

G L L

Λ =

Lift

d l

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ME 437/537-Particle

Lift

2 4 ) A L ( L

d 576 . F γ ρ =

−

&

2 / 3 3 2 / 1 ) Saff ( L

d 807 . F γ ρν = &

Leighton and Leighton and Acrivos Acrivos (1985) (1985)

Saffman Saffman

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ME 437/537-Particle

Velocity Field in the Inertial Velocity Field in the Inertial Sublayer Sublayer

B y ln 1 u + κ =

+ +

ν =

+

y u y

*

5 B ≈

300 y 30 ≤ <

+

Wall Units Wall Units

*

u u u =

+

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SLIDE 13

13

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dy dU µ = τ

dy dU u

2 *

ν =

1 dy dU =

+ +

+ + = y

u

5 y ≤ <

+

Turbulent stress is negligible Turbulent stress is negligible

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ME 437/537-Particle

+

y

+

u

5 . 5 y ln 5 . 2 u + =

+ + + + = y

u

30 12 300 10 20 30

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ME 437/537-Particle

ν = γ

2 *

u

4 ) A L ( L

d 576 . F

+ + −

=

3 ) Saff ( L

d 807 . F

+ +

=

2 L L

F F ρν =

+

ν =

+ *

du d

31 . 2 ) Hall ( L

d 21 . 4 F

+ +

=

87 . 1 ) MN ( L

d 57 . 15 F

+ +

=

Hall (1988) Hall (1988)

Mollinger Mollinger and and Nieuwstadt Nieuwstadt (1996) (1996)

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ME 437/537-Particle

1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Fl+

0.1 1 10

d+

Mollinger Hall Saffman Leighton Experiment