By P. Vainshtein and M. Shapiro Technion - Israel Institute of Technology Faculty of Mechanical...

by

P. Vainshtein and M. Shapiro

Technion - Israel Institute of TechnologyFaculty of Mechanical Engineering

An acoustic channel for aerosol particle focusing

Technion - Israel Institute of TechnologyFaculty of Mechanical Engineering 2

Background

Particles in gases are focused in aerodynamic lens arrays (P. McMurry and co-authors)

Ions are focused in quadrupole electrodynamic lenses(W. Paul and co-authors,1955)

Particles focusing in liquids by ultrasonic waves(Goddard, Martin, Graves, Kaduchak, 2006)


Aerosol

Aerodynamic lens arrays. Liu et al. (1995)

ParticleBeam

particle r0=1 cmgas

To vacuum chambers


Disadvantages of aerodynamic lens arrays:

- particles mix back into the flow as a result of transition to turbulence loss of particles, reduced instrument resolution.

- Use of small orifices (required to reduce gas flow rate and pumping capacity) is problematic because particle passages are blocked

- No control over beam broadening caused by particle diffusion poor focusing of nanoparticles

-No possibility of focusing at low gas velocities and constant gas pressures limits possibilities of implementation


Our solution

• Use acoustic field for focusing. Advantages: • No particles loss (characteristic of aerodynamic lenses)• Possibility to use at atmospheric pressures• No need of high gas velocity• No beam broadening due to side drift of nonspherical particles• Control over diffusion broadening• Improve transmission efficiency (transmit more particles)• When used in mass-spectrometers - improved sensitivity and

resolution


Physical principles of acoustic focusing

• Need narrow channels (small ) (larger acoustic effect, required in micro- nano-particle applications)•Need low acoustic frequencies to reduce attenuation •Therefore acoustic wave length no velocity nodes within the channel – no focusing?

• Examples

Pressure oscillations

Particles are either stagnant or move to the wall

Solution: use convex shaped channel (W. Paul, Electrodynamic quadrupole channel)

02r>>λ

02r


Quadrupole acoustic channel

x

Aerosol particle flow

Focused particles’ beam

tppp s ωcos0 =−

tppp s ωcos0 −=−

L


Streamlines and trajectories in quadrupole channel cross-section

Channel axis is the velocity node


022

rc>>=

ωπλ

Model

Condition of flow incompressibility

Small amplitude of pressure oscillations 10

<<=ppsε

Incompressible Navier-Stokes Eqs., creeping flow

Boundary conditions: quadrupole acoustic pressure disturbances at channel walls

0

2

=∇

∇+−∇=∂∂

v

vv νpt


( ) const,cos 22

20

s0 =

∂

∂−+

∂

∂=−

x

pzy

r

tp

x

pxpp

ω

Solution of Laplace equation for pressure

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

∂∂=

20

2

20

220 114

/

r

rU

r

rxpru

fνρ tr

zw

tr

y

s

s

ω

ω

sinv

,sinvv

0

0

=

−=

Fluid velocity

Axial componentCross-sectional components

wallschannelatacpx

pL <<∂∂


Equations of particle motion

,dt

dBr

pp Auuu

+−

=τ

)1(d

d

d

d 22

2

rt

x

t

xU −Π=+ωτ

tyt

y

t

ysin

d

d

d

d2

2

βωτ −=+

tzt

z

t

zsin

d

d

d

d2

2

βωτ =+2

00 )(2

v

r

p

r f

ss

ωρωβ ==

0r

UU ω=Π

00 =x

00 zy =initial particle velocities coincide with those of air

Dimensionless parameters:

axial fluid velocity

ωτ Stokes number

acoustic strength

Mathieu equations

Non-diffusive particles


10-1 100 101 1020

0.5

1

1.5

2

2.5

3

3.5

4

4.5

unstable

stable

Frequency parameter, ωτ

Acoustic strength parameter,

β

First marginal stability curve

a=1µmr0=0.5 cm

f=1 kHzSPL=140dB

1.0=ωτ47.0=β

stability

0.4

3.16


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45U=5cm/s,r0=0.5cm,SPL=140dB,f=1kHz,a=1μm

y

z

Calculated particle cross-sectional trajectories


10-4 10-3 10-2 10-1 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18U=5cm/s,r0=0.5cm,SPL=140dB,f=1kHz,a=1μmU=5cm/s,r0=0.5cm,SPL=140dB,f=1kHz,a=1μm

x

y

Calculated particle axial trajectories


10-1 100 101

40

60

80

100

120

140

160

SPL=140dB,f=1kHz

Frequency paarmeter, ωτ

t 1/10 ωτ

.

Dimensionless time of 10-fold focusing


0 100 200 300 400 500 600 700 800 900 1000

-0.1

0

0.1

0.2

0.3

0.4

a=25nm,U=5cm/sec,r0=5mm,f=1kHz,SPL=140dB

x

y

Diffusive low inertia particles 1<<ωτ


Achievable focusing width

systematic velocity

ru ps22

2

1 τβω−=

Velocity balance yields achievable focusing width

τωβ 2

21 DyB =

rDtDpB /2/v ==

200 )(

2v

r

p

r f

ss

ωρωβ ==

random velocity for estimate

Recall acoustic strength


Conclusions

• Practically interesting values of the frequency parameter and acoustic strength parameter lie well in the stability region• There exists critical value determining the maximal focusing efficiency of non-diffusive particles•Acoustic oscillations can focus micron size particles on axial distance comparable to channel cross-sectional size. •The achievable focusing width of small diffusive particles can be small enough. It is determined by the balance of random and systematic (acoustic) particle velocities.

ωτβ

1=ωτ

)1( <<ωτ

By P. Vainshtein and M. Shapiro Technion - Israel Institute of Technology Faculty of Mechanical...

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