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A Numerical Characterization of Particle Beam Collimation by an Aerodynamic Lens-Nozzle System: Part I. An Individual Lens or Nozzle

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Aerosol Science and Technology 36: 617
–
631 (2002)
c
°
2002 American Association for Aerosol ResearchPublished by Taylor and Francis0278-6826
=
02
=
$12.00
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A Numerical Characterization of Particle BeamCollimation by an Aerodynamic Lens-Nozzle System:Part I. An Individual Lens or Nozzle
Xuefeng Zhang,
1
Kenneth A. Smith,
1
Douglas R. Worsnop,
2
Jose Jimenez,
2
John T. Jayne,
2
and Charles E. Kolb
2
1
Department of Chemical Engineering, MassachusettsInstitute of Technology, Cambridge, Massachusetts
2
Center for Aerosol and Cloud Chemistry, Aerodyne Research, Inc, Billerica, Massachusetts
Particle beams have traditionally been produced by supersonicexpansionofaparticle-ladengasthroughasinglenozzletovacuum.However, it has been shown that, by passing the particle-laden gasthrough a series of axi-symmetric subsonic contractions and ex-pansions (an aerodynamic lens system) prior to the supersonic ex-pansion to vacuum through a single nozzle, beam divergence canbe signicantly reduced. In this paper, particle motion in expan-sions of a gas-particle suspension through either a single lens or asingle nozzle have been investigated numerically. Since the singleaerodynamic lens and the isolated nozzle are the elementary com-ponents of any aerodynamic lens-nozzle inlet system, a fundamen-tal understanding of these components is essential for designingan inlet system with the desired sampling rate, collimation, andtransmission properties. If a gas undergoes subsonic contractionand expansion through an orice, the associated particles wouldfollow the uid streamlines if the particles were inertialess. How-ever, real particles may either experience a displacement towardthe axis of symmetry or may impact on the front surface of thelens. The rst of these effects leads to collimation of the particlesnear the axis, but the second effect leads to particle loss. It is foundthatthe maximumparticledisplacementoccursataparticleStokesnumber, St, near unity and signicant impact loss also begins atSt
=
1. The lens dimensionless geometry and the Reynolds numberof the ow are other important parameters. When a gas contain-ing suspended particles undergoes supersonic expansion througha nozzle to vacuum from the lens working pressure (
300 Pa),it is found that particle beam divergence is a function of Reynoldsnumber,nozzlegeometry,andparticleStokesnumber.Morespecif-ically, it is found that a stepped nozzle generally helps to reducebeam divergence and that particle velocity scales with the speed of sound.
Received 15 January 2001; accepted 23 May 2001.This work was supported by EPA grant 82539-01-1.Address correspondence to Kenneth A. Smith, Departmentof Chemical Engineering, Massachusetts Institute of Technology,Cambridge, MA 02139-4307. E-mail: kas@eagle.mit.edu
INTRODUCTION
Particle beams have been used extensively in aerosol mea-surements since they were rst produced by Murphy and Sears(1964).Interestacceleratedinthelastdecadebecausecollimatedparticle beams have facilitated online measurements of the sizeand chemical composition of individual particles (e.g., Davis1977; Johnston and Wexler 1995; Murphy and Thomson 1995;Noble and Prather 1996; Jayne et al. 2000; and Tobias et al.2000). Such measurements are essential to a determination of the srcin of atmospheric particles and their impact on publichealth (Henry 1998).Almost all online particle sizing and chemical analysis tech-niques employ particle beams of controlled dimensions and di-vergences.Inthesetechniques,theparticlebeamisgeneratedbyexpanding a gas-particle suspension through single or multipleorices. Typically, particle time-of-ight (TOF) over a certaindistanceismeasuredbylightscatteringtodetermineparticleter-minalvelocity,fromwhichparticlesizecanbe inferred.Thepar-ticles are either subsequently ablated by a laser beam followedby TOF mass spectrometric analysis (Murphy and Thompson1995; Noble and Prather 1996) or are evaporated on a hot sur-facefollowedbyelectronimpactionizationandquadrupolemassspectrometric analysis (Davis 1977; Jayne et al. 2000; Tobiasetal.2000).RecentworkofJayneetal.(2000)hasdemonstratedthe ability to ascertain molecular composition of size-resolvedparticles by using a chopped particle beam followed by massspectrometricanalysis.Allofthesemeasurementsrequirehighlycollimated particle beams, especially for the laser-based tech-niques because the laser beams are highly focused (
0.5 mm indiameter, Mallina et al. (2000) Thomson et al. (1997)). Ideally,the particle beam should be fully located within the laser beam.Boththeoreticalandexperimentalapproacheshavebeenusedto investigate the factors that control beam diameter and diver-gence. Traditionally, the particle beam was generated by ex-panding a gas-particle suspension from atmospheric pressure tovacuum through a single orice,a method similar tothat used in617
618
X. ZHANG ET AL.
generating supersonic molecular beams (Kantrowitz and Grey1951).Considerablemeasurement and modeling effort has beendirected at providing quantitative information and fundamentalunderstanding of the properties of such particle beams. Israeland Friedlander (1967) generated a particle beam with a di-vergence angle of 0.0055 rad for the 126
–
365 nm particle sizerange using capillaries and a low pressure source (
100 torr).Note that the value of the beam divergence angle here is de-ned as the beam radius at a certain distance from the nozzleexit divided by the distance. Estes et al. (1983) characterizedcapillary-generated particle beams and found that the beam wasgenerally highly divergent (
>
0.02 rad divergence angle), ex-cept in a narrow range of particle diameters around 0.5 micron,for which the divergence angle was 0.005 rad. Dahneke andCheng (1979a,b) calculated performance characteristics of par-ticlebeamsgeneratedbyconicallyconvergentnozzlesandfoundthat if a detector collects particles within a 1
±
(0.017 rad) diver-gence angle, it will detect 0.5
–
1 micron particles with 100%efciency. Recent modeling results by Mallina et al. (1999) forconically convergent nozzles and for capillaries also found thatonlythose particleswithinanarrow sizerange(referredtoasthemaximal collimation diameter) are efciently collimated and,even for those maximally collimated particles, the divergenceangle was on the order of 0.008 rad. If one considers the max-imum beam divergence acceptable for current particle measur-ing instruments, these single nozzle systems are not adequate.For example, if one considers a sampling angle of 0.005 rad,which represents a 2.4 mm diameter resistance-heated particlevaporizer located 240 mm downstream, dimensions similar tothat used by Jayne et al. (2000), the above data suggest thatthe single nozzle systems rarely provide 100% collection ef-ciency even at the maximum collimation particle diameter. Forthose systems employing laser beams as size detectors or ion-ization sources, the sampling efciency is expected to be muchlower as the laser beams are generally focused to a diameterof
0.5 mm.To reduce beam divergence, sheath ow has been used toconne the particles to a region close to the axis prior to theexpansion through a single orice/nozzle to vacuum. Dahnekeand Cheng (1979b) investigated the effect of sheath ow on par-ticle beam divergence and found that beam divergence anglescan be reduced by a factor of 5 for 0.5 and 1.19 micron diam-eter polystyrene particles if 99% sheath ow is employed. Raoet al. (1993) and Kievit et al. (1996) also found that sheathow improves beam quality. However, sheath ow also hassome drawbacks, such as reducing particle sampling rate anddifculty in handling sheath gas, as pointed out by Liu et al.(1995a, b)and Kievitetal.(1996). Although itwasproposedbyDahneke and Flachsbart (1972) in the early 1970s, it is no sur-prise that sheath ow technology has rarely been used in thepractical particle analyses cited above. Alternatively, Liu et al.(1995a, b) centralized particles to a region close to the axis byforcing the gas-particle suspension to ow through a series of orices (referred to as an aerodynamic lens system) prior to ex-panding the gas to vacuum through a nal nozzle. It was foundthat the aerodynamic lens system provides the same functionas the sheath ow without reducing the particle sampling rateor creating complications in gas handling. The pioneering work by Liu et al. (1995a, b) and recent characterizations by Jayneetal.(2000)have demonstrated thatthetechnology signicantlyreduces beam divergence. The work of Schreiner et al. (1998,1999) has extended the application of aerodynamic lenses tocontexts in which the ambient pressure is low, such as in strato-spheric research.The aerodynamic lens-nozzle system proposed by Liu et al.(1995a, b) is shown schematically in Figure 1, which illustrateshow such a series of individual lenses can be used to effectivelytransportparticlesintovacuum.Theparticletrajectoriesthroughthe lens are calculated by Fluent (described later) for 500 nmdiameter spheres, which have a density of 1000 kg/m
3
. Exceptfor the trace closest tothe axis, each line inthe gure representsthe boundary of a region enclosing 10% of the particle owrate. The calculation assumes that the particles are disperseduniformly in the upstream gas and that the upstream gas veloc-ity prole is parabolic. A 2.4 mm diameter target (detector) islocated 240 mm downstream, and this conguration denes asampling angle of 5
£
10
¡
3
rad. Note that X
D
0 is placed at thenozzle exit for this particular gure. One can see that the beamis highly collimated with a divergence angle of about 10
¡
3
radso that the collection efciency on the target is 100%. The rst5 thin cylindrical orices serve to collimate the particles ontothe centerline; the nal exit orice generates a supersonic gasexpansion governing particle acceleration into the vacuum sys-tem. During the nal expansion, particles acquire a distributionof terminal velocities that depends on particle diameter, withsmaller diameter particles accelerating to faster velocities andlarger diameter particles accelerating to slower velocities. Mea-surement of this particle terminal velocity allows instrumentsequipped with aerodynamic lens systems to determine particleaerodynamic diameter.Liu et al. (1995a, b) analyzed, both theoretically and experi-mentally, the particle beams produced by an aerodynamic lens-nozzleexpansion.Intheircalculations,ow eldsintheaerody-namic lenses were assumed to be incompressible or isentropic,whereasthesupersonicfree-jetexpansionthoughthenozzlewascalculatedbyusingaquasi
–
one-dimensionalapproximationandan empirical expression, much like the procedure of Dahnekeand Cheng (1979a, b). Such calculations provide valuable in-formation concerning the factors (number of lenses, lens/nozzlegeometry) that control both beam quality and particle terminalvelocity. However, Liu et al.’s analysis was limited to particleslocated near the axisand tosmall particles (
D
p
<
250 nm). Theauthors were not able to study the loss of larger particles as aresult of impact on system surfaces.We have calculated the gas-particle ow eld in an aero-dynamic lens-nozzle expansion using the Fluent (Fluent, Inc.,Lebanon, NH) computational uid dynamics (CFD) package.Specically,FLUENT4.5.2wasusedandthegridwasgenerated
A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS
619
Figure 1.
Schematic of aerodynamic lens system and nozzle inlet showing the calculated trajectories of 500 nm particles.
P
up
D
280 Pa,
P
down
D
0
:
1 Pa, Re
0
D
20.8,
Q
D
97
:
6 scc/min, and
OD
D
10 mm. The detection target diameter is 2.4 mm, whichtogether with its distance from the nozzle (240 mm) denes a detection angle of 0.005 rad.
T
up
is gas upstream temperature.by GEOMESH. Fluent can simulate the full range of continuumsub-,trans-,andsupersonicowsintheaerodynamiclens-nozzleinlet without major assumptions, such asconstant entropy or in-compressibility. The CPU time (HP9000, 450 MHz CPU) fora typical calculation of the low speed ow in a lens is about1 h. For high speed ow in a nozzle it is about 10 h. The ob- jective of this work is to provide a fundamental understandingof the factors that control the formation of a beam contain-ing particles with diameters from 5 to 10,000 nm. This paperpresents systematic results on the motion of particles through asingle subsonic lens and on a supersonic expansion through anozzle to vacuum. Since the isolated aerodynamic lens andthe nozzle are the elementary components of an aerodynamiclens-nozzle inlet system, a fundamental understanding of thesecomponents is essential for designing the whole inlet systemwith the desired sampling rate and beam divergence. Detailedmodeling results for an integrated aerodynamic lens-nozzle in-let system, such as that shown in Figure 1, will be presented inanother paper.
SINGLE LENS SYSTEM
Figure2 isaschematicof asinglelenssystemwith aplot of agas streamline and a trajectory line for a 500 nm diameter parti-cle. The gure shows that, far upstream of the lens, the selectedparticle follows the gas toward the lens at a radial coordinate of
R
pi
D
4 mm. As the gas approaches the lens, it is acceleratedradially inward and one can see that the particle overshoots thegas streamlines toward the axis. Downstream of the lens, theparticle moves radially outward, but only slightly, and then set-tles onto a streamline with a smaller radial coordinate,
R
po
. Thelens is of inlet diameter
ID
1
, outlet diameter
ID
2
, and thickness
L
. The lens is inserted into a tube of diameter
OD
. Note that
ID
1
and
ID
2
are generally not equal, although
ID
1
D
ID
2
inFigure 2. The gas upstream pressure is280 Pa, and the upstreamgas Reynolds number, Re
0
(based on
OD
), is 12.5. The contrac-tion ratio, which describes the change in the radial position of atest particle, is
´
c
D
R
po
R
pi
[1]
NUMERICAL METHOD
The gas ow eld was calculated in Fluent by assumingparticle-free ow. After the gas ow eld was calculated, parti-cle trajectories were calculated by integrating the particle mo-mentum equation. This procedure requires that particle/particleinteractions be negligible and that the particles have little or noinuence on gas ow. A major assumption in the gas ow eldcalculation is the presumption of continuum ow. The presentlens and nozzle are designed to work at
300 Pa, at which pres-sure the gas mean free path (
¸
) is on the order of 20 microns.As the gas mean free path is much smaller than the lens/nozzledimensions (
L
1 mm with
¸
/
L
0
:
01), ow inside thelens-nozzle system can, in fact, be treated as a continuum ow.However,welldownstreamofthe nozzle,wherepressureisonly0.1 Pa (measured in a typical system, see Jayne et al. (2000)),and the dimension of the vacuum chamber,
L
, is about 100 mm,
620
X. ZHANG ET AL.
Figure 2.
Schematic of a single thin cylindrical lens showing a gas streamline and the trajectory of a 500 nm particle.
Q
D
60 sec/min,
L
D
4 mm,
ID
1
D
ID
2
D
2 mm,
OD
D
10 mm, Re
0
D
12.5,
P
up
D
280 Pa,
R
pi
D
4 mm,
R
po
D
0
:
5 mm.the ow is not of the continuum type as
¸
/
1. The gas oweld calculation with Fluent will not be valid under these condi-tions. However, the calculation shows that gas velocity reachesa maximum of about 5 mm downstream of the nozzle, at whichpoint the pressure is about 5 Pa. Here the calculation is stillvalid (
¸
D
1
:
2 mm, free jet
R
dimension
15 mm leading to
¸
/
L
0
:
1). Fortunately, the results show that events in the lowpressure region have little inuence on particle velocity exceptfor particles of
D
p
·
20 nm (5% reduction in terminal velocityhas been found for
D
p
D
20 nm particles during further ex-pansion to lower pressure). This conclusion is supported by thegood agreement between particle axial terminal velocity calcu-latedbyFluentandexperimentaldatadownto40nmindiameterby Jayne et al. (2000) and Liu et al. (1995b).Particles are assumed to be spherical throughout this work.However, it should be noted that the motion of nonsphericalparticles produces additional liftforcesso thatsuch particlesareless well collimated (Liu et al. 1995a, b; Jayne et al. 2000). Inthe calculation, gravity was neglected. However, for the largestparticle investigated (10 micron), it is estimated that the particledisplacement due to gravity within one lens spacing (50 mm) isabout 0.12 mm. This is marginally signicant and provides anincentivetokeep thelensspacingshortiflargeparticlesaretobemeasured. Alternatively, one could employ an instrument thatis vertically aligned. For smaller particles, the effect is entirelynegligible. Other assumptions are as follows: (a) perfect gas,(b) adiabatic ow, and (c) laminar ow. The Mach number forow in the tube and through the lens is very low, so the owcan be considered as either isothermal or adiabatic. The Machnumber in the nozzle is as high as 2 and the ow there is notisothermal, but it can be considered as adiabatic. The value of the Reynolds number in both the lens and nozzle is on the orderof 10, so the ow is laminar.To ensure that results on beam contraction are independentof the numerical grid density and the geometrical parameters of the computational domain, the calculations were subjected to aseries of tests. It will be shown later in the paper that the densityof the computation grid (5
–
15 grid/mm in
R
and 3
–
15 grid/mmin
X
)wasadequate,andthebordersofthecomputationaldomainup-anddownstreamofanoriceweremovedfarawayuntiltheydid not inuence the ow eld calculations signicantly.If the Mach number is low, as in the case of Figure 2 (M
0
0
:
03), the Reynolds number alone is sufcient to characterizethe gas ow eld:Re
0
D
½
0
V
0
OD
¹
0
;
[2]M
0
D
V
0
C
0
;
[3]where
V
0
,
¹
0
,
½
0
, and
C
0
are the average ow velocity, thegas viscosity, the gas density, and the sonic speed based on theupstream ow conditions.The equation of motion for a particle can be written as
dV
p
dt
D
V
¡
V
p
¿ ;
[4]where
V
p
is the particle velocity,
V
is the gas velocity, and
¿
isthe particlerelaxation time. If,asinthe currentcase,theparticleReynolds number (dened as
j
V
¡
V
p
j
½
g
D
p
=
¹
) is on the order
A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS
621of 0.01,
¿
can be written as
¿
D
½
p
D
p
2
C
s
18
¹
[5]where
½
p
and
D
p
are particledensity and diameter,respectively.
C
s
is a correction coefcient to Stokes’ law, which is importantifthe particle diameter isnot much larger than thegas mean freepath. It can be expressed as
C
s
D
1
C
Kn
p
[
A
C
Q
exp(
¡
B
=
Kn
p
)]
;
[6]where Kn
p
is the particle Knudsen number, dened as the ratioof the gas mean free path to particle diameter, and
A
,
Q
, and
B
are 1.21, 0.41, and 0.89, respectively (Allen and Raabe 1982).Typical values of Kn in this work were 2.2 to 4400, so particledrag is largely associated with the free molecular ow regime,for which
C
s
!
A
Kn
p
.Particle motion is usually characterized in terms of the Stokesnumber, which is dened asSt
D
¿
V
c
L
c
;
[7]where
V
c
and
L
c
are characteristic velocity and length scales,respectively.Equations (5)
–
(7), together with the well-known expressionfor mean free path, yieldsSt
D
8>><>>:
k
T
18
¹
p
2
¾
PV
c
L
c
½
p
D
p
;
free molecule regime
;
118
¹
V
c
L
c
½
p
D
2
p
;
continuum regime
;
[8]where
¾
is the molecular collision cross section,
P
is the gaspressure, and
k
is the Boltzmann constant. Equation (8) clearlyindicates that St in the free molecule regime is a function of gasmicroscopic properties (
¾
,
¹
, and
P
), gas velocity, lens char-acteristic dimension, and particle properties (
½
p
D
p
). Thereforefor agivenlenssystemoperating under constant owconditionsin the free molecule regime, the Stokes number depends on theproduct
½
p
D
p
.Inthecontinuumregime,however,thefunctionaldependence of particle Stokes number on the particle propertiesgoes as
½
p
D
p
2
.Notethatparticlediametersinthispaperarepre-sented as particle aerodynamic diameter (
½
p
D
p
with
½
p
equalto one g/cm
3
) unless specied.
RESULTS AND DISCUSSION
Particle Motion through a Single Lens
Numerical analysis of a single lens was undertaken in orderto develop an understanding of effects of lens geometry andow conditions on beam contraction. This understanding canprovide general guidance for the design of a lens system. For alldimensional results, it was assumed that the uid was air andthat the upstream temperature was 300 K.Figure 3 shows plots of particle trajectories through a sin-gle thin cylindrical lens with
ID
1
D
ID
2
D
5 mm, thickness
L
D
0
:
5 mm, and
OD
D
10 mm. In the single lens analy-sis, the upstream air pressure is 280 Pa and the ow rate,
Q
,is 60 sec/min. These parameters result in Re
0
D
12.5, whereRe
0
is based on the upstream quantities evaluated at
X
D
0.The top plot (Figure 3a) is for 5 nm diameter particles. It showsalmost no net radial displacement of the particles because theparticle inertia is so small that the particles closely followed thestreamlines. The middle plot (Figure 3b) is for 500 nm diameterparticles,and theparticletrajectories now exhibita strong radialcontraction effect. The bottom plot (Figure 3c) is for 10,000 nmdiameter particles. This plot shows that these particles have somuch inertia that their trajectories are hardly inuenced by thecontraction and expansion of the gas ow through the lens. An-other key feature in this plot is that some particles impact onthe lens, leading to particle loss and a reduction in transmis-sion efciency. In the present analysis of a single lens, particletransmission efciency,
´
t
,isdenedastheparticledownstreamow rate divided by the upstream ow rate. It is assumed thatparticles are uniformly dispersed in the air upstream of the lens
Figure 3.
Particletrajectoriesthrough asingle thin cylindricallens (
D
p
D
5,500, 1000 nm).
L
D
0
:
5 mm,
ID
1
D
ID
2
D
5 mm,
OD
D
10 mm,
Q
D
60 scc/min, Re
0
D
12.5, and
P
up
D
280 Pa.

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