1

Settling Velocity (Deposition)

Stokes' Law •  the drag on a spherical particle in a fluid is described by Stokes' Law for

the following conditions: –  fluid is a Newtonian incompressible fluid duk/dxk=0 –  gravity is negligible g=0 –  flow is creeping flow, i.e. Re<<1 duk/dxk=0 –  steady-state flow duj/dt=0

•  Navier-Stokes Equation –  Bird, Stewart and Lightfoot, 1960

•  for j=1, 2, 3... –  here we will neglect gravity so that the last term is dropped, and we will make

dimensionless using characteristic velocity u0 and length l •  where Re=ρul/µ=(inertial fluid forces)/(viscous fluid forces) •  l=diameter •  ρ=density •  µ=ρυ=fluid viscosity •  u=mean speed of the undisturbed flow upstream of the body

ρ∂u j

∂t+ uk

∂uj

∂xk

#

$ % &

' ( =∂p∂xj

+ µ∂ 2uj

∂xk∂xk

+ ρgj

ρ∂u j

*

∂t+ uk

* ∂u j*

∂xk*

#

$ % &

' ( =∂p*

∂xj* +

1Re

∂ 2uj*

∂xk*∂xk

*

Navier-Stokes Equation

•  Stokes' solution with the assumptions: –  –  infinite medium –  rigid sphere –  no slip at the surface of the

sphere •  so that in spherical

coordinates we get the following velocities

uk* ∂uk

*

∂xk* = 0

∂p*

∂xj* =

1Re

∂ 2uj*

∂xk*∂xk

*

ur = u∞ 1−

32

Rp

r#

$ % &

' +12

Rp

r#

$ % &

'

3(

) *

+

, - cosθ

uθ = u∞ 1 −

34

Rp

r$

% & '

( −14

Rp

r$

% & '

(

3)

* +

,

- . sinθ

p = p0 −

32

µu∞

Rp

#

$ %

&

' (

Rp

r#

$ % &

'

2

cosθ

´

Drag

•  drag force consists of 2 components –  normal force - pressure on the solid acting

perpendicularly to the surface at each point on the surface of the sphere; integrating around the sphere the normal force at any point (-pcosθ), the total normal force becomes

–  tangential force - shear stress caused by the velocity gradient in the vicinity of the surface

•  Fdrag=Fn+Ft –  if gravity≠0 then –  Ftotal=Fdrag+Fbuoyant –  if flow and gravity directions coincide then

•  for Re~1, inertial forces increase the drag force predicted by Stokes’ law

Fn = −p r =R p

cosθ( )0

π

∫0

2π

∫ Rp2 sinθdθdφ

= 2πµRpu∞

Ft = τ rθ r=Rpsinθ( )

0

π

∫0

2π

∫ Rp2 sinθdθdφ

= 4πµRpu∞ = 6πµRpu∞

Fbuoyant =

πDp3ρg6

•  examples of Reynolds numbers of particles of varied diameters in air at p=1 atm, T=20˚C.

•  for particles in any range of Reynolds number we can describe the drag force in terms of an empirical coefficient and particle projected area (Ap): –  any shape Fdrag=CDApρ(u2/2) –  spherical particles Fdrag=πCDDp

2ρu2/8

Dp (µm) Re 20 0.02 60 0.4 100 2 300 20

•  with drag coefficient is given by CD=24/Re Re<0.1 =(24/Re)[1+3Re/16+9Re2ln(2Re)/160] 0.1<Re<2 =(24/Re)[1+0.15Re0.687] 2<Re<500 =0.44 {Stokes' Law} 500<Re<2x105

Eq. 9.31

2

•  force balance on the particle

•  let the applicable forces be gravity and drag –  assume Re<0.1 such that CD=24/

Re

•  characteristic relaxation time, τ –  time scale required for the

approach to steady motion

•  terminal velocity of the particle in this fluid, vt, where the particle has reached steady state

mp

dvdt

= Fii∑

mp

dvdt

=mpg +3πµDp

Cc

u − v( )

τ =

mpCc

3πµDp

τ

dvdt

+ v = u − τg

0.1

9.2x10-8

9.0x10-7

1.0

3.6x10-6

3.5x10-5

10.0

3.1x10-4

3.0x10-3

Dp (µm)

τ (sec)

vt (m sec-1)

for unit density spheres in air at 20oC

€

vt = −τg

Diffusivity

gas A (carrier fluid) molecule

particle

gas B (second vapor) molecule

•  binary diffusivity –  using the Chapman-

Enskog theory and the hard-sphere approximation

–  DAB = (λAB cA)(3π/32)(1+z) [cm2 sec-1]

cf. Eq. 9.12, 9.13

Settling Velocity

•  terminal velocity of the particle in this fluid, vt, where the particle has reached steady state

0.1

9.2x10-8

9.0x10-7

1.0

3.6x10-6

3.5x10-5

10.0

3.1x10-4

3.0x10-3

Dp (µm)

τ (sec)

vt (m sec-1)

for unit density spheres in air at 20oC

Junge et al., 1961

Summary of Corrections to Stokes' Drag Force

Name

Drag coefficient

Cunningham correction factor or friction factor

Relative Magnitude

Factual > FStokes

Factual < FStokes

Range

Re=ρuDp/µ>0.1 Kn=2λ/Dp≥1

Applicable Values

Dp large, u large, ρ large, or µ small

Dp small or λ large

Definition

CD=Fdrag/(Apρ(u2/2))

Cc=3πµuDp/Fdrag

Factor

CD (non-creeping) [part. vel.]

Cc or f (non-continuum) [size]

Correction

CD=(24/Re), Re<0.1 CD=(24/Re)[1+3Re/16+9Re2ln(2Re)/160],

0.1<Re<2 CD=(24/Re)[1+0.15Re0.1687]. 2<Re<500

CD=0.44, 500<Re<2x105

Cc=1+Kn[1.257+0.4exp(-1.1/Kn)], or for air at STP

Cc=1+(1.257)2λ/Dp, Dp>>λ Cc=1+(1.657)2λ/Dp, Dp<<λ

Characteristic Length Scales

•  Knudsen number - ratio of the length scale of molecular motion in the fluid phase to the length scale of the particle; this ratio describes how the fluid "views" the particle, i.e. is the motion governed by the rules of molecules or of macroscopic objects –  Kn = 2λ/Dp= (fluid "graininess")/(particle radius)

•  Mean speed of gas molecules c c

c

3

Mean Free Path •  Mean free path of gases

–  mean free path - λair, the average distance traveled between collisions with other gas molecules; λAB is the average distance traveled by a molecule of A before it encounters a molecule of B (for Z collisions) λ = c/Z; c = (8kT/πm)0.5

•  Mean free path in particle evolution processes coagulation, deposition: diffusion of particles in air, use λair condensation: diffusion of another gas (B) to a particle in air, use λAB

λ λ

Eq. 9.3, 9.87

Regimes of Particle Motion

•  continuum regime – Kn << 1

Dp exceeds λair , so air appears to the particle as a continuum, and the laws of continuum mechanics apply

•  transition regime – Kn ≈ 1 λair and Dp are of the same order of magnitude, so transport is controlled by both continuum mechanics and kinetic theory

•  free molecule regime – Kn >> 1 λair exceeds Dp, so transport controlled by the kinetic theory of gases

λair

Dp

λair

½Dp

λair

λair

½Dp Dp

λair

λair

½Dp Dp

Slip Correction

•  continuum regime – Kn << 1 (Kn<0.1) No slip condition holds

•  transition regime – Kn ≈ 1 (0.1<Kn<10) Slip correction is required

•  free molecule regime – Kn >> 1 (10<Kn) Drag force is smaller than predicted by Stokes

λair

½Dp

λair

½Dp

λair

λair

½Dp ½Dp

λair

λair

½Dp ½Dp

Cunningham Correction Factor

•  continuum regime – Kn << 1 (Kn<0.1) No slip condition holds

•  transition regime – Kn ≈ 1 (0.1<Kn<10) Slip correction is required

•  free molecule regime – Kn >> 1 (10<Kn) Drag force is smaller than predicted by Stokes

Dp (µm)

Cc

10.0 1.016

1.0 1.164

0.1 2.867

0.01 22.218

Junge and Gustaffson, 1957 Junge et al., 1961

How spherical are real

particles?

4

Particle Size and Deposition

•  Sizes

– Particle Size Distributions (defining “size”) – How to Calculate Mean Particle Size

•  Microphysics – Deposition Velocity (depends on size) – How to Calculate Particle Lifetime – Bonus: How to Calculate Particle Loss in a Tube

Aerosol Composition

•  Chemical composition gives an indication of particle sources

•  C, N, S contributions to composition illustrate role of aerosols in biogeochemical cycles

Classification of Pollutants

•  Fine Particles –  less than 2.5 µm

in diameter

•  Coarse Particles –  greater than 2.5

µm in diameter

Particle Types and Sizes

0.0001 0.001 0.01 0.1 1 10

Particle size (microns)

Bacteria

Black Carbon (Soot)

Tobacco Smoke

Viruses

Gas Molecules

Size Range for Particle Sources

Particle Type Size Range automotive emissions 0.01 µm to 1 µm

bacteria 0.2 µm to 10+ µm

tobacco smoke 0.01 µm to 1 µm

viruses 0.002 µm to 0.05 µm

Particle Types and their Removal

Flagan and Seinfeld, 1986