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04/22/2023 1Aerosol & Particulate Research Laboratory
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