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Single Particle MotionReading: Chaps 3 & 5

Want to be great athletes? Study aerosol science & engineering because you surely need to know how to control particle movement in the air!

04/22/2023 2Aerosol & Particulate Research Laboratory

• Newton’s Resistance Law The force is proportional to the gas pushed away and the relative velocity between the sphere and the gas (negligible viscous force)

Vddtdm

pg2

4

22

8VdCF pgDD

CD = 0.44 (sphere) for 103 < Rep < 2×105

· Reynolds Number: ratio of inertial forces to frictional forces (Chap.2.5)

L

FF g

f

Ip

VRe

For a spherical particle, L = dp,

V

Delivery van 0.04Sports car 1.0Airplane 0.25

Q: Choose a combination of dp and V when the Newton’s Law can be applied.

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• Stokes’ Law: negligible inertial force compared to viscous force (Rep < 1); in a laminar flow for a spherical particle

F VdD p3

pDC

Re24

Assumptions used in solving Navier Stokes’ equation: • Incompressible flow (g = cons)• Constant motion (V = const)• Rigid sphere• Fluid velocity on the particle surface

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Settling Velocity• When the drag force is equal and opposite to the

gravitational force

• Particle Mechanical Mobility

• Settling Velocity

F F mg

Vdd g

D G

pp g p

36

3

( )

1Re & 1for

18

2

md

gdV

p

ppTS

BVF d

d mD p

p 1

31

for

V F BTS G

Q: What is the physical meaning of B?Q: Does a smaller or a larger particle have larger mobility?

~0

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• Cunningham Slip Correction Factor: gas velocity at the surface of small particles is not zero --> slip

39.0exp05.134.21

)1.0( 52.21

3

p

pc

pp

c

c

pD

dd

C

mdd

C

CVd

F : mean free path (Chap 2.3)

0.066 m for air @STP

Q: So is the corrected VTS larger or smaller?

(Re < 1)

1 mm 0.1 m

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Aerodynamic Diameter• The Stokes diameter, ds, is the diameter of the sphere that

has the same density and settling velocity as the particle.• The aerodynamic diameter, da, is the diameter of the unit

density (0 = 1 g/cm3) sphere that has the same settling velocity as the particle.

1818 18

20

22 gdgdgdV assep

TS

00

22

0

ss

pea

s

e

s

p

a

ep

ddd

dd

dd

Q: Can we design an instrument by applying settling velocity?

Cunningham factor should be included if dp < 1 m

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LHVV

x

TS

Q: Can it be vertical?Q: Can we make the acceleration > g?

Horizontal Elutriator (settling chamber, spectrometer)

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Vertical Elutriator

http://getdomainvids.com

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Centrifuge Separator

Cheng et al., 1988

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Particle Acceleration• Newton’s law

)1()(

)()(

)()(3

/

tTS

pDG

eVtVdt

tdVtVgdt

tdVmdtVmgFF

FG=mg

t=0V(t)=0

FG=mg

t=V(t)=VTS

FG=mg

t>3V(t)=VTS

FD=3V(t)dp

FD=3VTSdp

= mB : relaxation time

Q: What is the physical meaning of relaxation time?

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Non-zero Initial velocityV t V V V ef f

t( ) ( ) / 0

Stopping Distance

1) Re(for 6

Rearctan6Re

1) Re(for

0

3/103/1

0

000

g

ppdS

BmVVS

ª An aerosol can adjust itself very quickly to a new environment!!!

Time for unit density particles to reach their terminal velocitydp (m) 3 (ms) S* (cm)

0.01 0.00002 6.810-6

0.1 0.00026 8.810-5

1 0.011 3.610-3

10 0.85 0.23100 65 12.7

* V0=1000 cm/s

/0 1)( t

ff eVVtVtx Displacement

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Inertial Impaction• Stokes number: the ratio of the stopping distance of a

particle to a characteristic dimension of the obstacle

• For an impactor

StkSd

Udc c

0

j

cpp

j

DUCd

DUStk

9

2/

2

f(Stk) efficiencyImpaction

Q: Stk << 1? Stk >> 1?

http://aerosol.ees.ufl.edu/

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Impactor type Stk50 Stk50 Circular nozzle 0.24 0.49 Rectangular nozzle 0.59 0.77

500 < Re (nozzle throat) < 3000 and h'/Dj > 1.5

Stk50 for 2 impactors

Determine dp50

jetr rectangula 9

jet round 4

9

9

502

503

50

50

QLStkW

QStkD

UStkD

Cd

p

p

j

p

j

cp

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Cascade Impactor

Aerosol flow In

Clean air out

http://aerosol.ees.ufl.edu/instrumentation/section04.html

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Q: Advantages? Disadvantages?

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Virtual Impactor

Collection efficiency & internal loss curves for a virtual impactor

Aerosol concentrator? Cyclone?

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CYCLONE

VortexFinder

Cylinder

Cone

Inlet

Outlet

DustDischarger

Cyclonebody

http://aerosol.ees.ufl.edu

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Time of Flight

Q: Comparison with an impactor?

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Aerodynamic Focusing Lens

TSI Operating Manual, 2004

Time of Flight (TOF) vs Time in Beam (TIB)

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Quick Reflection