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

SLIDE 1

1

Ahmadi ME 637 Ahmadi ME 637

! ! Introduction to Aerosols Introduction to Aerosols ! ! Drag Forces Drag Forces ! ! Cunningham Corrections Cunningham Corrections ! ! Lift Forces Lift Forces

Ahmadi ME 637

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. .

Ahmadi ME 637

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 × ×

SLIDE 2

2

Ahmadi ME 637

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

Ahmadi ME 637

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 vp

p = Particle Velocity

= Particle Velocity v v’ ’ = Thermal Velocity = Thermal Velocity v vf

f = Fluid (Air) Velocity

= Fluid (Air) Velocity n = Number Density n = Number Density c = Speed of Sound c = Speed of Sound

Ahmadi ME 637

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

Ahmadi ME 637

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

SLIDE 3

3

Ahmadi ME 637

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

Ahmadi ME 637

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

Ahmadi ME 637

D

C

Re

Ahmadi ME 637

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

SLIDE 4

4

Ahmadi ME 637

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.

Ahmadi ME 637

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

Ahmadi ME 637

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.

Ahmadi ME 637

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

SLIDE 5

5

Ahmadi ME 637

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

Ahmadi ME 637

K=Correction Factor K=Correction Factor

K Ud 3 F

e D

πµ =

3 / 1 e

) Volume 6 ( d π =

Ahmadi ME 637

Drag Gravity

g u u u

p f p

m ) ( C d 3 dt d m

c

+ − πµ =

Equation of Motion Equation of Motion

Ahmadi ME 637

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

−

SLIDE 6

6

Ahmadi ME 637

) 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

Ahmadi ME 637

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

Ahmadi ME 637

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

Ahmadi ME 637

)] 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

SLIDE 7

7

Ahmadi ME 637

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.

Ahmadi ME 637

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.

Ahmadi ME 637

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

Ahmadi ME 637

) 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

SLIDE 8

8

Ahmadi ME 637

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)

Ahmadi ME 637

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)

Ahmadi ME 637

⎩ ⎨ ⎧ << 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)

Ahmadi ME 637

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

SLIDE 9

9

Ahmadi ME 637

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)

Ahmadi ME 637

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

Ahmadi ME 637

! ! Introduction to Aerosols Introduction to Aerosols ! ! Drag Forces Drag Forces ! ! Cunningham Corrections Cunningham Corrections ! ! Lift Forces Lift Forces

Ahmadi ME 637