Aerosol Science and Technology 36 617ndash631 (2002)cdeg 2002 American Association for Aerosol ResearchPublished by Taylor and Francis0278-6826=02=$1200 C 00

A Numerical Characterization of Particle BeamCollimation by an Aerodynamic Lens-Nozzle SystemPart I An Individual Lens or Nozzle

Xuefeng Zhang1 Kenneth A Smith1 Douglas R Worsnop2 Jose Jimenez2

John T Jayne2 and Charles E Kolb2

1Department of Chemical Engineering Massachusetts Institute of Technology Cambridge Massachusetts2Center for Aerosol and Cloud Chemistry Aerodyne Research Inc Billerica Massachusetts

Particle beams have traditionally been produced by supersonicexpansion of a particle-laden gas through a single nozzle to vacuumHowever 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 signi cantly 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 ori ce 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 foundthat the maximum particle displacement occurs at a particle Stokesnumber St near unity and signi cant impact loss also begins atSt raquo= 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 (raquo 300 Pa)it is found that particle beam divergence is a function of Reynoldsnumber nozzle geometry and particle Stokes number More specif-ically it is found that a stepped nozzle generally helps to reducebeam divergence and that particle velocity scales with the speed ofsound

Received 15 January 2001 accepted 23 May 2001This work was supported by EPA grant 82539-01-1Address correspondence to Kenneth A Smith Department

of Chemical Engineering Massachusetts Institute of TechnologyCambridge MA 02139-4307 E-mail kaseaglemitedu

INTRODUCTIONParticle beams have been used extensively in aerosol mea-

surements since they were rst produced by Murphy and Sears(1964) Interest accelerated in the last decade because collimatedparticle beams have facilitated online measurements of the sizeand chemical composition of individual particles (eg Davis1977 Johnston and Wexler 1995 Murphy and Thomson 1995Noble and Prather 1996 Jayne et al 2000 and Tobias et al2000) Such measurements are essential to a determination ofthe origin 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 In these techniques the particle beam is generated byexpanding a gas-particle suspension through single or multipleori ces Typically particle time-of- ight (TOF) over a certaindistance is measured by light scattering to determine particle ter-minal velocity from which particle size can be inferred The par-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-face followedby electron impact ionization and quadrupole massspectrometric analysis (Davis 1977 Jayne et al 2000 Tobiaset al 2000) Recent work of Jayne et al (2000) has demonstratedthe ability to ascertain molecular composition of size-resolvedparticles by using a chopped particle beam followed by massspectrometric analysis All of these measurements require highlycollimated particle beams especially for the laser-based tech-niques because the laser beams are highly focused (raquo05 mm indiameter Mallina et al (2000) Thomson et al (1997)) Ideallythe particle beam should be fully located within the laser beam

Both theoretical and experimental approaches have been usedto 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 ori ce a method similar to that used in

617

618 X ZHANG ET AL

generating supersonic molecular beams (Kantrowitz and Grey1951) Considerable measurement 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 00055 rad for the 126ndash365 nm particle sizerange using capillaries and a low pressure source (raquo100 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 (gt002 rad divergence angle) ex-cept in a narrow range of particle diameters around 05 micronfor which the divergence angle was 0005 rad Dahneke andCheng (1979ab) calculated performance characteristics of par-ticle beams generated by conically convergent nozzles and foundthat if a detector collects particles within a 1plusmn (0017 rad) diver-gence angle it will detect 05ndash1 micron particles with 100ef ciency Recent modeling results by Mallina et al (1999) forconically convergent nozzles and for capillaries also found thatonly those particles within a narrow size range (referred to as themaximal collimation diameter) are ef ciently collimated andeven for those maximally collimated particles the divergenceangle was on the order of 0008 rad If one considers the max-imum beam divergence acceptable for current particle measur-ing instruments these single nozzle systems are not adequateFor example if one considers a sampling angle of 0005 radwhich represents a 24 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 ef ciency is expected to be muchlower as the laser beams are generally focused to a diameterof raquo05 mm

To reduce beam divergence sheath ow has been used tocon ne the particles to a region close to the axis prior to theexpansion through a single ori cenozzle 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 05 and 119 micron diam-eter polystyrene particles if 99 sheath ow is employed Raoet al (1993) and Kievit et al (1996) also found that sheath ow improves beam quality However sheath ow also hassome drawbacks such as reducing particle sampling rate anddif culty in handling sheath gas as pointed out by Liu et al(1995a b) and Kievit et al (1996) Although it was proposed byDahneke 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 ofori ces (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 workby Liu et al (1995a b) and recent characterizations by Jayneet al (2000) have demonstrated that the technology signi cantlyreduces beam divergence The work of Schreiner et al (19981999) 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 effectivelytransport particles into vacuum The particle trajectories throughthe lens are calculated by Fluent (described later) for 500 nmdiameter spheres which have a density of 1000 kgm3 Exceptfor the trace closest to the axis each line in the 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 pro le is parabolic A 24 mm diameter target (detector) islocated 240 mm downstream and this con guration de nes asampling angle of 5 pound 10iexcl3 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 10iexcl3 radso that the collection ef ciency on the target is 100 The rst5 thin cylindrical ori ces serve to collimate the particles ontothe centerline the nal exit ori ce 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-nozzle expansion In their calculations ow elds in the aerody-namic lenses were assumed to be incompressible or isentropicwhereas the supersonic free-jet expansion though the nozzle wascalculated by using a quasindashone-dimensional approximation andan empirical expression much like the procedure of Dahnekeand Cheng (1979a b) Such calculations provide valuable in-formation concerning the factors (number of lenses lensnozzlegeometry) that control both beam quality and particle terminalvelocity However Liu et alrsquos analysis was limited to particleslocated near the axis and to small particles (Dp lt 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 IncLebanon NH) computational uid dynamics (CFD) packageSpeci cally FLUENT452 was used and the grid was generated

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 particlesPup D 280 Pa Pdown D 01 Pa Re0 D 208 Q D 976 sccmin and OD D 10 mm The detection target diameter is 24 mm whichtogether with its distance from the nozzle (240 mm) de nes a detection angle of 0005 rad Tup is gas upstream temperature

by GEOMESH Fluent can simulate the full range of continuumsub- trans- and supersonic ows in the aerodynamic lens-nozzleinlet without major assumptions such as constant 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 10000 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 SYSTEMFigure 2 is a schematic of a single lens system with a plot of a

gas 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 ofRpi 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 the

particle moves radially outward but only slightly and then set-tles onto a streamline with a smaller radial coordinate Rpo Thelens is of inlet diameter ID1 outlet diameter ID2 and thicknessL The lens is inserted into a tube of diameter OD Note thatID1 and ID2 are generally not equal although ID1 D ID2 inFigure 2 The gas upstream pressure is 280 Pa and the upstreamgas Reynolds number Re0 (based on OD) is 125 The contrac-tion ratio which describes the change in the radial position of atest particle is

acutec DRpo

Rpi[1]

NUMERICAL METHODThe gas ow eld was calculated in Fluent by assuming

particle-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 particleparticleinteractions be negligible and that the particles have little or noin uence 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 raquo300 Pa at which pres-sure the gas mean free path (cedil) is on the order of 20 micronsAs the gas mean free path is much smaller than the lensnozzledimensions (L raquo 1 mm with cedilL raquo 001) ow inside thelens-nozzle system can in fact be treated as a continuum owHowever well downstream of the nozzle where pressure is only01 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 D60 secmin L D 4 mm ID1 D ID2 D 2 mm OD D 10 mm Re0 D 125 Pup D 280 Pa Rpi D 4 mm Rpo D 05 mm

the ow is not of the continuum type as cedilraquo1 The gas ow eld 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 (cedil D 12 mm free jet R dimension raquo15 mm leading tocedilL raquo 01) Fortunately the results show that events in the lowpressure region have little in uence on particle velocity exceptfor particles of Dp middot 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-lated by Fluent and experimental data down to 40 nm in diameterby Jayne et al (2000) and Liu et al (1995b)

Particles are assumed to be spherical throughout this workHowever it should be noted that the motion of nonsphericalparticles produces additional lift forces so that such particles areless 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 012 mm This is marginally signi cant and provides anincentive to keep the lens spacing short if large particles are to bemeasured 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 for ow 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 ofthe computational domain the calculations were subjected to aseries of tests It will be shown later in the paper that the densityof the computation grid (5ndash15 gridmm in R and 3ndash15 gridmmin X) was adequate and the borders of the computational domainup- and downstream of an ori ce were moved far away until theydid not in uence the ow eld calculations signi cantly

If the Mach number is low as in the case of Figure 2 (M0 raquo003) the Reynolds number alone is suf cient to characterizethe gas ow eld

Re0 Dfrac120V0OD

sup10 [2]

M0 DV0

C0 [3]

where V0 sup10 frac120 and C0 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

dtD

V iexcl Vp

iquest [4]

where Vp is the particle velocity V is the gas velocity and iquest isthe particle relaxation time If as in the current case the particleReynolds number (de ned as jV iexcl Vpjfrac12g Dp=sup1) is on the order

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 621

of 001 iquest can be written as

iquest Dfrac12p D p

2Cs

18sup1[5]

where frac12p and D p are particle density and diameter respectivelyCs is a correction coef cient to Stokesrsquo law which is importantif the particle diameter is not much larger than the gas mean freepath It can be expressed as

Cs D 1 C Knp[A C Q exp(iexclB=Knp )] [6]

where Knp is the particle Knudsen number de ned as the ratioof the gas mean free path to particle diameter and A Q and Bare 121 041 and 089 respectively (Allen and Raabe 1982)Typical values of Kn in this work were 22 to 4400 so particledrag is largely associated with the free molecular ow regimefor which Cs AKnpParticle motion is usually characterized in terms of the Stokesnumber which is de ned as

St Diquest Vc

L c [7]

where Vc and L c are characteristic velocity and length scalesrespectively

Equations (5)ndash(7) together with the well-known expressionfor mean free path yields

St D

8gtgtlt

gtgt

kT

18sup1p

2frac34 P

Vc

Lcfrac12p Dp free molecule regime

1

18sup1

Vc

Lcfrac12p D2

p continuum regime

[8]

where frac34 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 (frac34 sup1 and P) gas velocity lens char-acteristic dimension and particle properties (frac12pDp ) Thereforefor a given lens system operating under constant ow conditionsin the free molecule regime the Stokes number depends on theproduct frac12pDp In the continuum regime however the functionaldependence of particle Stokes number on the particle propertiesgoes as frac12p D p

2 Note that particle diameters in this paper are pre-sented as particle aerodynamic diameter (frac12pDp with frac12p equalto one gcm3) unless speci ed

RESULTS AND DISCUSSION

Particle Motion through a Single LensNumerical analysis of a single lens was undertaken in order

to develop an understanding of effects of lens geometry and ow conditions on beam contraction This understanding canprovide general guidance for the design of a lens system For all

dimensional 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 ID1 D ID2 D 5 mm thicknessL D 05 mm and OD D 10 mm In the single lens analy-sis the upstream air pressure is 280 Pa and the ow rate Qis 60 secmin These parameters result in Re0 D 125 whereRe0 is based on the upstream quantities evaluated at X D 0The 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 the particle trajectories now exhibit a strong radialcontraction effect The bottom plot (Figure 3c) is for 10000 nmdiameter particles This plot shows that these particles have somuch inertia that their trajectories are hardly in uenced 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 ef ciency In the present analysis of a single lens particletransmission ef ciency acutet is de ned as the particle downstream ow rate divided by the upstream ow rate It is assumed thatparticles are uniformly dispersed in the air upstream of the lens

Figure 3 Particle trajectories through a single thin cylindricallens (D p D 5500 1000 nm) L D 05 mm ID1 D ID2 D 5 mmOD D 10 mm Q D 60 sccmin Re0 D 125 and Pup D 280 Pa

622 X ZHANG ET AL

Figure 4 Particle downstream radial coordinate Rpo versus upstream radial coordinate Rpi for Dp D 5 500 and 10000 nmOther parameters are the same as in Figure 3

and that any particle that impacts on the ori ce plate is lost fromthe ow For a parabolic upstream velocity pro le and purelyballistic particles the transmission ef ciency is given by

acutet b D 2

sup3ID

OD

acute2

1 iexcl12

sup3ID

OD

acute2

[9]

This provides a lower bound for the actual transmissionef ciency

In Figure 4 the particle trajectory results from Figure 3 areplotted as downstream radial-coordinate Rpo versus upstreamradial-coordinate Rpi The results for Dp D 5 nm form a straightand nearly diagonal line which con rms the lack of any signif-icant radial contraction The line for D p D 500 nm is almoststraight with a slope of about 05 This indicates that the con-traction ratio acutec is a constant equal to 05 A roughly linearrelationship between Rpo and Rpi was also observed for D p D10000 nm but with a larger slope due to higher particle inertiaThe results in Figure 4 suggest that the beam contraction ratio isroughly constant It is therefore appropriate to use the contrac-tion ratio acutec D RpoRpi at a certain Rpi to quantify the lens colli-mation effect Throughout the present work acutec at Rpi D 15 mmor RpiOD D 015 is used to represent overall lens performanceThis is similar to the near axis acutec used by Liu et al (1995a b)

Figure 5 shows results for acutec versus particle diameter for atypical lens (ID D 5 mm OD D 10 mm L D 05 mm) oper-ating at a typical condition (Pup D 280 Pa Q D 60 secminRe0 D 125) but at different grid densities One can see that eachcurve in Figure 5a starts from acutec close to unity because particleinertia is too small to produce a contraction and then reaches aminimum at a certain diameter which is referred to as the maxi-

mal collimation diameter Finally acutec increases for particles thatare too heavy to respond to the gas contractionexpansion Thissuggests that for a given lens upstream pressure particles of di-ameters around the maximal collimation diameter are ef cientlydisplaced Figure 5a demonstrates that double or half of the nor-mal grid density (R-11 gridmm X-7 gridmm squares) doesnot signi cantly alter the results (maximum deviation is about1) but using one quarter of the normal grid density produces anoticeable deviation (6) from the calculation with the normalgrid density

Figure 5b provides corresponding results for the transmissionef ciency (acutet ) versus particle diameter The particle transmis-sion ef ciency starts at unity because there is no impact lossfor small particles At a characteristic diameter at which par-ticles have appreciable displacement in the top plot there is asharp drop in the transmission ef ciency due to impact loss Fi-nally for suf ciently large particles the impact loss is simplygeometrical blocking ie particles at Rpi gt ID2 are blockedby the lens whereas all others are transmitted as described byEquation (9) Similar to the results for acutec plotted in Figure 5aFigure 5b demonstrates that double or half of the normal griddensity (R-11 gridmm X-7 gridmm) does not signi cantly al-ter the results for acutet (maximum deviation is about 25) but onequarter of the normal grid density results in clear deviation (6)from the calculation with the normal grid density It should benoted that the accuracy of acutet is also in uenced by the resolutionin Rpi (currently 01 mm) for test particles For instance for thecircles and squares at 104 nm in Figure 5b it was found that testparticles injected Rip D 25 mm passed through the lens but testparticles at Rip D 26 mm impacted acutet is calculated as 0438based on Rpi D 25 mm as the maximum radius for particles

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 623

Figure 5 Particle beam contraction ratio (a) and transmissionef ciency(b)versus particle diameter for a single thin cylindricallens calculated with different grid densities Parameters are thesame as in Figure 3

that pass through the lens The triangles are the cases for whichRpi D 26 mm particles pass through the lens but Rpi D 27 mmparticles did not and this lead to acutet D 0468 This introduces amaximum uncertainty of 6 and no effort to increase the Rpi

resolution has been attemptedFor the low speed expansion of a dilute gas-particle mixture

through a lens as shown in Figure 3 the gas exhibits little com-pressibility so the contraction ratio (acutec ) and the transmissionef ciency (acutet ) must be functions only of lens geometry gas up-stream Re0 (which describes the gas ow eld) and particle St(which describes the departure of the particle trajectory from thegas streamline) One would also expect that maximum contrac-tion and signi cant particle impact loss would both occur at Staround 1 For the case of a cylindrical lens the relationships canbe expressed as

acutec D f

sup3ID

OD

L

ID Re0 St

acute [10]

acutet D g

sup3ID

OD

L

ID Re0 St

acute [11]

where ID D ID1 D ID2

To test the validity of Equations (10)ndash(11) acutec and acutet werecalculated by changing the ow eld parameters with lens di-mensions xed or changing lens dimensions with ow eld pa-rameters xed Re0 and the dimensionless lens geometry(IDOD) were kept constant in either case Operationally fora lens of xed geometry acutec and acutet were calculated by specify-ing a number of different upstream pressures For each upstreampressure a trial-and-error calculation was then undertaken to de-termine that value of the downstream pressure that would givethe same gas mass ow rate as had been found in the base caseThis is equivalent to keeping the upstream Reynolds numberconstant Results of this calculation for a thin plate lens areplotted versus particle diameter in Figure 6 where upstreamgas pressures were 140ndash340 Pa whereas lens geometry and gasmass ow rate are unchanged (ID D 5 mm OD D 10 mmL D 05 mm and Q D 60 sccmin Re0 D 125) One cansee that each curve in Figure 6 exhibits trends that are sim-ilar to those in Figure 5 In addition it is observed that themaximal collimation particle size is smaller for lower upstream

Figure 6 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle diameter for a single thin ori celens with gas upstream pressure varying from 140 Pa to 340 PaRe0 D 125 ID D 5 mm OD D 10 mm L D 05 mm Q D60 sccmin

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

618 X ZHANG ET AL

generating supersonic molecular beams (Kantrowitz and Grey1951) Considerable measurement 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 00055 rad for the 126ndash365 nm particle sizerange using capillaries and a low pressure source (raquo100 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 (gt002 rad divergence angle) ex-cept in a narrow range of particle diameters around 05 micronfor which the divergence angle was 0005 rad Dahneke andCheng (1979ab) calculated performance characteristics of par-ticle beams generated by conically convergent nozzles and foundthat if a detector collects particles within a 1plusmn (0017 rad) diver-gence angle it will detect 05ndash1 micron particles with 100ef ciency Recent modeling results by Mallina et al (1999) forconically convergent nozzles and for capillaries also found thatonly those particles within a narrow size range (referred to as themaximal collimation diameter) are ef ciently collimated andeven for those maximally collimated particles the divergenceangle was on the order of 0008 rad If one considers the max-imum beam divergence acceptable for current particle measur-ing instruments these single nozzle systems are not adequateFor example if one considers a sampling angle of 0005 radwhich represents a 24 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 ef ciency is expected to be muchlower as the laser beams are generally focused to a diameterof raquo05 mm

To reduce beam divergence sheath ow has been used tocon ne the particles to a region close to the axis prior to theexpansion through a single ori cenozzle 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 05 and 119 micron diam-eter polystyrene particles if 99 sheath ow is employed Raoet al (1993) and Kievit et al (1996) also found that sheath ow improves beam quality However sheath ow also hassome drawbacks such as reducing particle sampling rate anddif culty in handling sheath gas as pointed out by Liu et al(1995a b) and Kievit et al (1996) Although it was proposed byDahneke 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 ofori ces (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 workby Liu et al (1995a b) and recent characterizations by Jayneet al (2000) have demonstrated that the technology signi cantlyreduces beam divergence The work of Schreiner et al (19981999) 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 effectivelytransport particles into vacuum The particle trajectories throughthe lens are calculated by Fluent (described later) for 500 nmdiameter spheres which have a density of 1000 kgm3 Exceptfor the trace closest to the axis each line in the 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 pro le is parabolic A 24 mm diameter target (detector) islocated 240 mm downstream and this con guration de nes asampling angle of 5 pound 10iexcl3 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 10iexcl3 radso that the collection ef ciency on the target is 100 The rst5 thin cylindrical ori ces serve to collimate the particles ontothe centerline the nal exit ori ce 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-nozzle expansion In their calculations ow elds in the aerody-namic lenses were assumed to be incompressible or isentropicwhereas the supersonic free-jet expansion though the nozzle wascalculated by using a quasindashone-dimensional approximation andan empirical expression much like the procedure of Dahnekeand Cheng (1979a b) Such calculations provide valuable in-formation concerning the factors (number of lenses lensnozzlegeometry) that control both beam quality and particle terminalvelocity However Liu et alrsquos analysis was limited to particleslocated near the axis and to small particles (Dp lt 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 IncLebanon NH) computational uid dynamics (CFD) packageSpeci cally FLUENT452 was used and the grid was generated

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 particlesPup D 280 Pa Pdown D 01 Pa Re0 D 208 Q D 976 sccmin and OD D 10 mm The detection target diameter is 24 mm whichtogether with its distance from the nozzle (240 mm) de nes a detection angle of 0005 rad Tup is gas upstream temperature

by GEOMESH Fluent can simulate the full range of continuumsub- trans- and supersonic ows in the aerodynamic lens-nozzleinlet without major assumptions such as constant 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 10000 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 SYSTEMFigure 2 is a schematic of a single lens system with a plot of a

gas 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 ofRpi 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 the

particle moves radially outward but only slightly and then set-tles onto a streamline with a smaller radial coordinate Rpo Thelens is of inlet diameter ID1 outlet diameter ID2 and thicknessL The lens is inserted into a tube of diameter OD Note thatID1 and ID2 are generally not equal although ID1 D ID2 inFigure 2 The gas upstream pressure is 280 Pa and the upstreamgas Reynolds number Re0 (based on OD) is 125 The contrac-tion ratio which describes the change in the radial position of atest particle is

acutec DRpo

Rpi[1]

NUMERICAL METHODThe gas ow eld was calculated in Fluent by assuming

particle-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 particleparticleinteractions be negligible and that the particles have little or noin uence 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 raquo300 Pa at which pres-sure the gas mean free path (cedil) is on the order of 20 micronsAs the gas mean free path is much smaller than the lensnozzledimensions (L raquo 1 mm with cedilL raquo 001) ow inside thelens-nozzle system can in fact be treated as a continuum owHowever well downstream of the nozzle where pressure is only01 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 D60 secmin L D 4 mm ID1 D ID2 D 2 mm OD D 10 mm Re0 D 125 Pup D 280 Pa Rpi D 4 mm Rpo D 05 mm

the ow is not of the continuum type as cedilraquo1 The gas ow eld 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 (cedil D 12 mm free jet R dimension raquo15 mm leading tocedilL raquo 01) Fortunately the results show that events in the lowpressure region have little in uence on particle velocity exceptfor particles of Dp middot 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-lated by Fluent and experimental data down to 40 nm in diameterby Jayne et al (2000) and Liu et al (1995b)

Particles are assumed to be spherical throughout this workHowever it should be noted that the motion of nonsphericalparticles produces additional lift forces so that such particles areless 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 012 mm This is marginally signi cant and provides anincentive to keep the lens spacing short if large particles are to bemeasured 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 for ow 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 ofthe computational domain the calculations were subjected to aseries of tests It will be shown later in the paper that the densityof the computation grid (5ndash15 gridmm in R and 3ndash15 gridmmin X) was adequate and the borders of the computational domainup- and downstream of an ori ce were moved far away until theydid not in uence the ow eld calculations signi cantly

If the Mach number is low as in the case of Figure 2 (M0 raquo003) the Reynolds number alone is suf cient to characterizethe gas ow eld

Re0 Dfrac120V0OD

sup10 [2]

M0 DV0

C0 [3]

where V0 sup10 frac120 and C0 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

dtD

V iexcl Vp

iquest [4]

where Vp is the particle velocity V is the gas velocity and iquest isthe particle relaxation time If as in the current case the particleReynolds number (de ned as jV iexcl Vpjfrac12g Dp=sup1) is on the order

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 621

of 001 iquest can be written as

iquest Dfrac12p D p

2Cs

18sup1[5]

where frac12p and D p are particle density and diameter respectivelyCs is a correction coef cient to Stokesrsquo law which is importantif the particle diameter is not much larger than the gas mean freepath It can be expressed as

Cs D 1 C Knp[A C Q exp(iexclB=Knp )] [6]

where Knp is the particle Knudsen number de ned as the ratioof the gas mean free path to particle diameter and A Q and Bare 121 041 and 089 respectively (Allen and Raabe 1982)Typical values of Kn in this work were 22 to 4400 so particledrag is largely associated with the free molecular ow regimefor which Cs AKnpParticle motion is usually characterized in terms of the Stokesnumber which is de ned as

St Diquest Vc

L c [7]

where Vc and L c are characteristic velocity and length scalesrespectively

Equations (5)ndash(7) together with the well-known expressionfor mean free path yields

St D

8gtgtlt

gtgt

kT

18sup1p

2frac34 P

Vc

Lcfrac12p Dp free molecule regime

1

18sup1

Vc

Lcfrac12p D2

p continuum regime

[8]

where frac34 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 (frac34 sup1 and P) gas velocity lens char-acteristic dimension and particle properties (frac12pDp ) Thereforefor a given lens system operating under constant ow conditionsin the free molecule regime the Stokes number depends on theproduct frac12pDp In the continuum regime however the functionaldependence of particle Stokes number on the particle propertiesgoes as frac12p D p

2 Note that particle diameters in this paper are pre-sented as particle aerodynamic diameter (frac12pDp with frac12p equalto one gcm3) unless speci ed

RESULTS AND DISCUSSION

Particle Motion through a Single LensNumerical analysis of a single lens was undertaken in order

to develop an understanding of effects of lens geometry and ow conditions on beam contraction This understanding canprovide general guidance for the design of a lens system For all

dimensional 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 ID1 D ID2 D 5 mm thicknessL D 05 mm and OD D 10 mm In the single lens analy-sis the upstream air pressure is 280 Pa and the ow rate Qis 60 secmin These parameters result in Re0 D 125 whereRe0 is based on the upstream quantities evaluated at X D 0The 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 the particle trajectories now exhibit a strong radialcontraction effect The bottom plot (Figure 3c) is for 10000 nmdiameter particles This plot shows that these particles have somuch inertia that their trajectories are hardly in uenced 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 ef ciency In the present analysis of a single lens particletransmission ef ciency acutet is de ned as the particle downstream ow rate divided by the upstream ow rate It is assumed thatparticles are uniformly dispersed in the air upstream of the lens

Figure 3 Particle trajectories through a single thin cylindricallens (D p D 5500 1000 nm) L D 05 mm ID1 D ID2 D 5 mmOD D 10 mm Q D 60 sccmin Re0 D 125 and Pup D 280 Pa

622 X ZHANG ET AL

Figure 4 Particle downstream radial coordinate Rpo versus upstream radial coordinate Rpi for Dp D 5 500 and 10000 nmOther parameters are the same as in Figure 3

and that any particle that impacts on the ori ce plate is lost fromthe ow For a parabolic upstream velocity pro le and purelyballistic particles the transmission ef ciency is given by

acutet b D 2

sup3ID

OD

acute2

1 iexcl12

sup3ID

OD

acute2

[9]

This provides a lower bound for the actual transmissionef ciency

In Figure 4 the particle trajectory results from Figure 3 areplotted as downstream radial-coordinate Rpo versus upstreamradial-coordinate Rpi The results for Dp D 5 nm form a straightand nearly diagonal line which con rms the lack of any signif-icant radial contraction The line for D p D 500 nm is almoststraight with a slope of about 05 This indicates that the con-traction ratio acutec is a constant equal to 05 A roughly linearrelationship between Rpo and Rpi was also observed for D p D10000 nm but with a larger slope due to higher particle inertiaThe results in Figure 4 suggest that the beam contraction ratio isroughly constant It is therefore appropriate to use the contrac-tion ratio acutec D RpoRpi at a certain Rpi to quantify the lens colli-mation effect Throughout the present work acutec at Rpi D 15 mmor RpiOD D 015 is used to represent overall lens performanceThis is similar to the near axis acutec used by Liu et al (1995a b)

Figure 5 shows results for acutec versus particle diameter for atypical lens (ID D 5 mm OD D 10 mm L D 05 mm) oper-ating at a typical condition (Pup D 280 Pa Q D 60 secminRe0 D 125) but at different grid densities One can see that eachcurve in Figure 5a starts from acutec close to unity because particleinertia is too small to produce a contraction and then reaches aminimum at a certain diameter which is referred to as the maxi-

mal collimation diameter Finally acutec increases for particles thatare too heavy to respond to the gas contractionexpansion Thissuggests that for a given lens upstream pressure particles of di-ameters around the maximal collimation diameter are ef cientlydisplaced Figure 5a demonstrates that double or half of the nor-mal grid density (R-11 gridmm X-7 gridmm squares) doesnot signi cantly alter the results (maximum deviation is about1) but using one quarter of the normal grid density produces anoticeable deviation (6) from the calculation with the normalgrid density

Figure 5b provides corresponding results for the transmissionef ciency (acutet ) versus particle diameter The particle transmis-sion ef ciency starts at unity because there is no impact lossfor small particles At a characteristic diameter at which par-ticles have appreciable displacement in the top plot there is asharp drop in the transmission ef ciency due to impact loss Fi-nally for suf ciently large particles the impact loss is simplygeometrical blocking ie particles at Rpi gt ID2 are blockedby the lens whereas all others are transmitted as described byEquation (9) Similar to the results for acutec plotted in Figure 5aFigure 5b demonstrates that double or half of the normal griddensity (R-11 gridmm X-7 gridmm) does not signi cantly al-ter the results for acutet (maximum deviation is about 25) but onequarter of the normal grid density results in clear deviation (6)from the calculation with the normal grid density It should benoted that the accuracy of acutet is also in uenced by the resolutionin Rpi (currently 01 mm) for test particles For instance for thecircles and squares at 104 nm in Figure 5b it was found that testparticles injected Rip D 25 mm passed through the lens but testparticles at Rip D 26 mm impacted acutet is calculated as 0438based on Rpi D 25 mm as the maximum radius for particles

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 623

Figure 5 Particle beam contraction ratio (a) and transmissionef ciency(b)versus particle diameter for a single thin cylindricallens calculated with different grid densities Parameters are thesame as in Figure 3

that pass through the lens The triangles are the cases for whichRpi D 26 mm particles pass through the lens but Rpi D 27 mmparticles did not and this lead to acutet D 0468 This introduces amaximum uncertainty of 6 and no effort to increase the Rpi

resolution has been attemptedFor the low speed expansion of a dilute gas-particle mixture

through a lens as shown in Figure 3 the gas exhibits little com-pressibility so the contraction ratio (acutec ) and the transmissionef ciency (acutet ) must be functions only of lens geometry gas up-stream Re0 (which describes the gas ow eld) and particle St(which describes the departure of the particle trajectory from thegas streamline) One would also expect that maximum contrac-tion and signi cant particle impact loss would both occur at Staround 1 For the case of a cylindrical lens the relationships canbe expressed as

acutec D f

sup3ID

OD

L

ID Re0 St

acute [10]

acutet D g

sup3ID

OD

L

ID Re0 St

acute [11]

where ID D ID1 D ID2

To test the validity of Equations (10)ndash(11) acutec and acutet werecalculated by changing the ow eld parameters with lens di-mensions xed or changing lens dimensions with ow eld pa-rameters xed Re0 and the dimensionless lens geometry(IDOD) were kept constant in either case Operationally fora lens of xed geometry acutec and acutet were calculated by specify-ing a number of different upstream pressures For each upstreampressure a trial-and-error calculation was then undertaken to de-termine that value of the downstream pressure that would givethe same gas mass ow rate as had been found in the base caseThis is equivalent to keeping the upstream Reynolds numberconstant Results of this calculation for a thin plate lens areplotted versus particle diameter in Figure 6 where upstreamgas pressures were 140ndash340 Pa whereas lens geometry and gasmass ow rate are unchanged (ID D 5 mm OD D 10 mmL D 05 mm and Q D 60 sccmin Re0 D 125) One cansee that each curve in Figure 6 exhibits trends that are sim-ilar to those in Figure 5 In addition it is observed that themaximal collimation particle size is smaller for lower upstream

Figure 6 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle diameter for a single thin ori celens with gas upstream pressure varying from 140 Pa to 340 PaRe0 D 125 ID D 5 mm OD D 10 mm L D 05 mm Q D60 sccmin

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

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 particlesPup D 280 Pa Pdown D 01 Pa Re0 D 208 Q D 976 sccmin and OD D 10 mm The detection target diameter is 24 mm whichtogether with its distance from the nozzle (240 mm) de nes a detection angle of 0005 rad Tup is gas upstream temperature

by GEOMESH Fluent can simulate the full range of continuumsub- trans- and supersonic ows in the aerodynamic lens-nozzleinlet without major assumptions such as constant 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 10000 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 SYSTEMFigure 2 is a schematic of a single lens system with a plot of a

gas 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 ofRpi 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 the

particle moves radially outward but only slightly and then set-tles onto a streamline with a smaller radial coordinate Rpo Thelens is of inlet diameter ID1 outlet diameter ID2 and thicknessL The lens is inserted into a tube of diameter OD Note thatID1 and ID2 are generally not equal although ID1 D ID2 inFigure 2 The gas upstream pressure is 280 Pa and the upstreamgas Reynolds number Re0 (based on OD) is 125 The contrac-tion ratio which describes the change in the radial position of atest particle is

acutec DRpo

Rpi[1]

NUMERICAL METHODThe gas ow eld was calculated in Fluent by assuming

particle-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 particleparticleinteractions be negligible and that the particles have little or noin uence 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 raquo300 Pa at which pres-sure the gas mean free path (cedil) is on the order of 20 micronsAs the gas mean free path is much smaller than the lensnozzledimensions (L raquo 1 mm with cedilL raquo 001) ow inside thelens-nozzle system can in fact be treated as a continuum owHowever well downstream of the nozzle where pressure is only01 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 D60 secmin L D 4 mm ID1 D ID2 D 2 mm OD D 10 mm Re0 D 125 Pup D 280 Pa Rpi D 4 mm Rpo D 05 mm

the ow is not of the continuum type as cedilraquo1 The gas ow eld 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 (cedil D 12 mm free jet R dimension raquo15 mm leading tocedilL raquo 01) Fortunately the results show that events in the lowpressure region have little in uence on particle velocity exceptfor particles of Dp middot 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-lated by Fluent and experimental data down to 40 nm in diameterby Jayne et al (2000) and Liu et al (1995b)

Particles are assumed to be spherical throughout this workHowever it should be noted that the motion of nonsphericalparticles produces additional lift forces so that such particles areless 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 012 mm This is marginally signi cant and provides anincentive to keep the lens spacing short if large particles are to bemeasured 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 for ow 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 ofthe computational domain the calculations were subjected to aseries of tests It will be shown later in the paper that the densityof the computation grid (5ndash15 gridmm in R and 3ndash15 gridmmin X) was adequate and the borders of the computational domainup- and downstream of an ori ce were moved far away until theydid not in uence the ow eld calculations signi cantly

If the Mach number is low as in the case of Figure 2 (M0 raquo003) the Reynolds number alone is suf cient to characterizethe gas ow eld

Re0 Dfrac120V0OD

sup10 [2]

M0 DV0

C0 [3]

where V0 sup10 frac120 and C0 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

dtD

V iexcl Vp

iquest [4]

where Vp is the particle velocity V is the gas velocity and iquest isthe particle relaxation time If as in the current case the particleReynolds number (de ned as jV iexcl Vpjfrac12g Dp=sup1) is on the order

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 621

of 001 iquest can be written as

iquest Dfrac12p D p

2Cs

18sup1[5]

where frac12p and D p are particle density and diameter respectivelyCs is a correction coef cient to Stokesrsquo law which is importantif the particle diameter is not much larger than the gas mean freepath It can be expressed as

Cs D 1 C Knp[A C Q exp(iexclB=Knp )] [6]

where Knp is the particle Knudsen number de ned as the ratioof the gas mean free path to particle diameter and A Q and Bare 121 041 and 089 respectively (Allen and Raabe 1982)Typical values of Kn in this work were 22 to 4400 so particledrag is largely associated with the free molecular ow regimefor which Cs AKnpParticle motion is usually characterized in terms of the Stokesnumber which is de ned as

St Diquest Vc

L c [7]

where Vc and L c are characteristic velocity and length scalesrespectively

Equations (5)ndash(7) together with the well-known expressionfor mean free path yields

St D

8gtgtlt

gtgt

kT

18sup1p

2frac34 P

Vc

Lcfrac12p Dp free molecule regime

1

18sup1

Vc

Lcfrac12p D2

p continuum regime

[8]

where frac34 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 (frac34 sup1 and P) gas velocity lens char-acteristic dimension and particle properties (frac12pDp ) Thereforefor a given lens system operating under constant ow conditionsin the free molecule regime the Stokes number depends on theproduct frac12pDp In the continuum regime however the functionaldependence of particle Stokes number on the particle propertiesgoes as frac12p D p

2 Note that particle diameters in this paper are pre-sented as particle aerodynamic diameter (frac12pDp with frac12p equalto one gcm3) unless speci ed

RESULTS AND DISCUSSION

Particle Motion through a Single LensNumerical analysis of a single lens was undertaken in order

to develop an understanding of effects of lens geometry and ow conditions on beam contraction This understanding canprovide general guidance for the design of a lens system For all

dimensional 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 ID1 D ID2 D 5 mm thicknessL D 05 mm and OD D 10 mm In the single lens analy-sis the upstream air pressure is 280 Pa and the ow rate Qis 60 secmin These parameters result in Re0 D 125 whereRe0 is based on the upstream quantities evaluated at X D 0The 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 the particle trajectories now exhibit a strong radialcontraction effect The bottom plot (Figure 3c) is for 10000 nmdiameter particles This plot shows that these particles have somuch inertia that their trajectories are hardly in uenced 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 ef ciency In the present analysis of a single lens particletransmission ef ciency acutet is de ned as the particle downstream ow rate divided by the upstream ow rate It is assumed thatparticles are uniformly dispersed in the air upstream of the lens

Figure 3 Particle trajectories through a single thin cylindricallens (D p D 5500 1000 nm) L D 05 mm ID1 D ID2 D 5 mmOD D 10 mm Q D 60 sccmin Re0 D 125 and Pup D 280 Pa

622 X ZHANG ET AL

Figure 4 Particle downstream radial coordinate Rpo versus upstream radial coordinate Rpi for Dp D 5 500 and 10000 nmOther parameters are the same as in Figure 3

and that any particle that impacts on the ori ce plate is lost fromthe ow For a parabolic upstream velocity pro le and purelyballistic particles the transmission ef ciency is given by

acutet b D 2

sup3ID

OD

acute2

1 iexcl12

sup3ID

OD

acute2

[9]

This provides a lower bound for the actual transmissionef ciency

In Figure 4 the particle trajectory results from Figure 3 areplotted as downstream radial-coordinate Rpo versus upstreamradial-coordinate Rpi The results for Dp D 5 nm form a straightand nearly diagonal line which con rms the lack of any signif-icant radial contraction The line for D p D 500 nm is almoststraight with a slope of about 05 This indicates that the con-traction ratio acutec is a constant equal to 05 A roughly linearrelationship between Rpo and Rpi was also observed for D p D10000 nm but with a larger slope due to higher particle inertiaThe results in Figure 4 suggest that the beam contraction ratio isroughly constant It is therefore appropriate to use the contrac-tion ratio acutec D RpoRpi at a certain Rpi to quantify the lens colli-mation effect Throughout the present work acutec at Rpi D 15 mmor RpiOD D 015 is used to represent overall lens performanceThis is similar to the near axis acutec used by Liu et al (1995a b)

Figure 5 shows results for acutec versus particle diameter for atypical lens (ID D 5 mm OD D 10 mm L D 05 mm) oper-ating at a typical condition (Pup D 280 Pa Q D 60 secminRe0 D 125) but at different grid densities One can see that eachcurve in Figure 5a starts from acutec close to unity because particleinertia is too small to produce a contraction and then reaches aminimum at a certain diameter which is referred to as the maxi-

mal collimation diameter Finally acutec increases for particles thatare too heavy to respond to the gas contractionexpansion Thissuggests that for a given lens upstream pressure particles of di-ameters around the maximal collimation diameter are ef cientlydisplaced Figure 5a demonstrates that double or half of the nor-mal grid density (R-11 gridmm X-7 gridmm squares) doesnot signi cantly alter the results (maximum deviation is about1) but using one quarter of the normal grid density produces anoticeable deviation (6) from the calculation with the normalgrid density

Figure 5b provides corresponding results for the transmissionef ciency (acutet ) versus particle diameter The particle transmis-sion ef ciency starts at unity because there is no impact lossfor small particles At a characteristic diameter at which par-ticles have appreciable displacement in the top plot there is asharp drop in the transmission ef ciency due to impact loss Fi-nally for suf ciently large particles the impact loss is simplygeometrical blocking ie particles at Rpi gt ID2 are blockedby the lens whereas all others are transmitted as described byEquation (9) Similar to the results for acutec plotted in Figure 5aFigure 5b demonstrates that double or half of the normal griddensity (R-11 gridmm X-7 gridmm) does not signi cantly al-ter the results for acutet (maximum deviation is about 25) but onequarter of the normal grid density results in clear deviation (6)from the calculation with the normal grid density It should benoted that the accuracy of acutet is also in uenced by the resolutionin Rpi (currently 01 mm) for test particles For instance for thecircles and squares at 104 nm in Figure 5b it was found that testparticles injected Rip D 25 mm passed through the lens but testparticles at Rip D 26 mm impacted acutet is calculated as 0438based on Rpi D 25 mm as the maximum radius for particles

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 623

Figure 5 Particle beam contraction ratio (a) and transmissionef ciency(b)versus particle diameter for a single thin cylindricallens calculated with different grid densities Parameters are thesame as in Figure 3

that pass through the lens The triangles are the cases for whichRpi D 26 mm particles pass through the lens but Rpi D 27 mmparticles did not and this lead to acutet D 0468 This introduces amaximum uncertainty of 6 and no effort to increase the Rpi

resolution has been attemptedFor the low speed expansion of a dilute gas-particle mixture

through a lens as shown in Figure 3 the gas exhibits little com-pressibility so the contraction ratio (acutec ) and the transmissionef ciency (acutet ) must be functions only of lens geometry gas up-stream Re0 (which describes the gas ow eld) and particle St(which describes the departure of the particle trajectory from thegas streamline) One would also expect that maximum contrac-tion and signi cant particle impact loss would both occur at Staround 1 For the case of a cylindrical lens the relationships canbe expressed as

acutec D f

sup3ID

OD

L

ID Re0 St

acute [10]

acutet D g

sup3ID

OD

L

ID Re0 St

acute [11]

where ID D ID1 D ID2

To test the validity of Equations (10)ndash(11) acutec and acutet werecalculated by changing the ow eld parameters with lens di-mensions xed or changing lens dimensions with ow eld pa-rameters xed Re0 and the dimensionless lens geometry(IDOD) were kept constant in either case Operationally fora lens of xed geometry acutec and acutet were calculated by specify-ing a number of different upstream pressures For each upstreampressure a trial-and-error calculation was then undertaken to de-termine that value of the downstream pressure that would givethe same gas mass ow rate as had been found in the base caseThis is equivalent to keeping the upstream Reynolds numberconstant Results of this calculation for a thin plate lens areplotted versus particle diameter in Figure 6 where upstreamgas pressures were 140ndash340 Pa whereas lens geometry and gasmass ow rate are unchanged (ID D 5 mm OD D 10 mmL D 05 mm and Q D 60 sccmin Re0 D 125) One cansee that each curve in Figure 6 exhibits trends that are sim-ilar to those in Figure 5 In addition it is observed that themaximal collimation particle size is smaller for lower upstream

Figure 6 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle diameter for a single thin ori celens with gas upstream pressure varying from 140 Pa to 340 PaRe0 D 125 ID D 5 mm OD D 10 mm L D 05 mm Q D60 sccmin

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

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 D60 secmin L D 4 mm ID1 D ID2 D 2 mm OD D 10 mm Re0 D 125 Pup D 280 Pa Rpi D 4 mm Rpo D 05 mm

the ow is not of the continuum type as cedilraquo1 The gas ow eld 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 (cedil D 12 mm free jet R dimension raquo15 mm leading tocedilL raquo 01) Fortunately the results show that events in the lowpressure region have little in uence on particle velocity exceptfor particles of Dp middot 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-lated by Fluent and experimental data down to 40 nm in diameterby Jayne et al (2000) and Liu et al (1995b)

Particles are assumed to be spherical throughout this workHowever it should be noted that the motion of nonsphericalparticles produces additional lift forces so that such particles areless 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 012 mm This is marginally signi cant and provides anincentive to keep the lens spacing short if large particles are to bemeasured 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 for ow 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 ofthe computational domain the calculations were subjected to aseries of tests It will be shown later in the paper that the densityof the computation grid (5ndash15 gridmm in R and 3ndash15 gridmmin X) was adequate and the borders of the computational domainup- and downstream of an ori ce were moved far away until theydid not in uence the ow eld calculations signi cantly

If the Mach number is low as in the case of Figure 2 (M0 raquo003) the Reynolds number alone is suf cient to characterizethe gas ow eld

Re0 Dfrac120V0OD

sup10 [2]

M0 DV0

C0 [3]

where V0 sup10 frac120 and C0 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

dtD

V iexcl Vp

iquest [4]

where Vp is the particle velocity V is the gas velocity and iquest isthe particle relaxation time If as in the current case the particleReynolds number (de ned as jV iexcl Vpjfrac12g Dp=sup1) is on the order

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 621

of 001 iquest can be written as

iquest Dfrac12p D p

2Cs

18sup1[5]

where frac12p and D p are particle density and diameter respectivelyCs is a correction coef cient to Stokesrsquo law which is importantif the particle diameter is not much larger than the gas mean freepath It can be expressed as

Cs D 1 C Knp[A C Q exp(iexclB=Knp )] [6]

where Knp is the particle Knudsen number de ned as the ratioof the gas mean free path to particle diameter and A Q and Bare 121 041 and 089 respectively (Allen and Raabe 1982)Typical values of Kn in this work were 22 to 4400 so particledrag is largely associated with the free molecular ow regimefor which Cs AKnpParticle motion is usually characterized in terms of the Stokesnumber which is de ned as

St Diquest Vc

L c [7]

where Vc and L c are characteristic velocity and length scalesrespectively

Equations (5)ndash(7) together with the well-known expressionfor mean free path yields

St D

8gtgtlt

gtgt

kT

18sup1p

2frac34 P

Vc

Lcfrac12p Dp free molecule regime

1

18sup1

Vc

Lcfrac12p D2

p continuum regime

[8]

where frac34 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 (frac34 sup1 and P) gas velocity lens char-acteristic dimension and particle properties (frac12pDp ) Thereforefor a given lens system operating under constant ow conditionsin the free molecule regime the Stokes number depends on theproduct frac12pDp In the continuum regime however the functionaldependence of particle Stokes number on the particle propertiesgoes as frac12p D p

2 Note that particle diameters in this paper are pre-sented as particle aerodynamic diameter (frac12pDp with frac12p equalto one gcm3) unless speci ed

RESULTS AND DISCUSSION

Particle Motion through a Single LensNumerical analysis of a single lens was undertaken in order

to develop an understanding of effects of lens geometry and ow conditions on beam contraction This understanding canprovide general guidance for the design of a lens system For all

dimensional 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 ID1 D ID2 D 5 mm thicknessL D 05 mm and OD D 10 mm In the single lens analy-sis the upstream air pressure is 280 Pa and the ow rate Qis 60 secmin These parameters result in Re0 D 125 whereRe0 is based on the upstream quantities evaluated at X D 0The 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 the particle trajectories now exhibit a strong radialcontraction effect The bottom plot (Figure 3c) is for 10000 nmdiameter particles This plot shows that these particles have somuch inertia that their trajectories are hardly in uenced 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 ef ciency In the present analysis of a single lens particletransmission ef ciency acutet is de ned as the particle downstream ow rate divided by the upstream ow rate It is assumed thatparticles are uniformly dispersed in the air upstream of the lens

Figure 3 Particle trajectories through a single thin cylindricallens (D p D 5500 1000 nm) L D 05 mm ID1 D ID2 D 5 mmOD D 10 mm Q D 60 sccmin Re0 D 125 and Pup D 280 Pa

622 X ZHANG ET AL

Figure 4 Particle downstream radial coordinate Rpo versus upstream radial coordinate Rpi for Dp D 5 500 and 10000 nmOther parameters are the same as in Figure 3

and that any particle that impacts on the ori ce plate is lost fromthe ow For a parabolic upstream velocity pro le and purelyballistic particles the transmission ef ciency is given by

acutet b D 2

sup3ID

OD

acute2

1 iexcl12

sup3ID

OD

acute2

[9]

This provides a lower bound for the actual transmissionef ciency

In Figure 4 the particle trajectory results from Figure 3 areplotted as downstream radial-coordinate Rpo versus upstreamradial-coordinate Rpi The results for Dp D 5 nm form a straightand nearly diagonal line which con rms the lack of any signif-icant radial contraction The line for D p D 500 nm is almoststraight with a slope of about 05 This indicates that the con-traction ratio acutec is a constant equal to 05 A roughly linearrelationship between Rpo and Rpi was also observed for D p D10000 nm but with a larger slope due to higher particle inertiaThe results in Figure 4 suggest that the beam contraction ratio isroughly constant It is therefore appropriate to use the contrac-tion ratio acutec D RpoRpi at a certain Rpi to quantify the lens colli-mation effect Throughout the present work acutec at Rpi D 15 mmor RpiOD D 015 is used to represent overall lens performanceThis is similar to the near axis acutec used by Liu et al (1995a b)

Figure 5 shows results for acutec versus particle diameter for atypical lens (ID D 5 mm OD D 10 mm L D 05 mm) oper-ating at a typical condition (Pup D 280 Pa Q D 60 secminRe0 D 125) but at different grid densities One can see that eachcurve in Figure 5a starts from acutec close to unity because particleinertia is too small to produce a contraction and then reaches aminimum at a certain diameter which is referred to as the maxi-

mal collimation diameter Finally acutec increases for particles thatare too heavy to respond to the gas contractionexpansion Thissuggests that for a given lens upstream pressure particles of di-ameters around the maximal collimation diameter are ef cientlydisplaced Figure 5a demonstrates that double or half of the nor-mal grid density (R-11 gridmm X-7 gridmm squares) doesnot signi cantly alter the results (maximum deviation is about1) but using one quarter of the normal grid density produces anoticeable deviation (6) from the calculation with the normalgrid density

Figure 5b provides corresponding results for the transmissionef ciency (acutet ) versus particle diameter The particle transmis-sion ef ciency starts at unity because there is no impact lossfor small particles At a characteristic diameter at which par-ticles have appreciable displacement in the top plot there is asharp drop in the transmission ef ciency due to impact loss Fi-nally for suf ciently large particles the impact loss is simplygeometrical blocking ie particles at Rpi gt ID2 are blockedby the lens whereas all others are transmitted as described byEquation (9) Similar to the results for acutec plotted in Figure 5aFigure 5b demonstrates that double or half of the normal griddensity (R-11 gridmm X-7 gridmm) does not signi cantly al-ter the results for acutet (maximum deviation is about 25) but onequarter of the normal grid density results in clear deviation (6)from the calculation with the normal grid density It should benoted that the accuracy of acutet is also in uenced by the resolutionin Rpi (currently 01 mm) for test particles For instance for thecircles and squares at 104 nm in Figure 5b it was found that testparticles injected Rip D 25 mm passed through the lens but testparticles at Rip D 26 mm impacted acutet is calculated as 0438based on Rpi D 25 mm as the maximum radius for particles

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 623

Figure 5 Particle beam contraction ratio (a) and transmissionef ciency(b)versus particle diameter for a single thin cylindricallens calculated with different grid densities Parameters are thesame as in Figure 3

that pass through the lens The triangles are the cases for whichRpi D 26 mm particles pass through the lens but Rpi D 27 mmparticles did not and this lead to acutet D 0468 This introduces amaximum uncertainty of 6 and no effort to increase the Rpi

resolution has been attemptedFor the low speed expansion of a dilute gas-particle mixture

through a lens as shown in Figure 3 the gas exhibits little com-pressibility so the contraction ratio (acutec ) and the transmissionef ciency (acutet ) must be functions only of lens geometry gas up-stream Re0 (which describes the gas ow eld) and particle St(which describes the departure of the particle trajectory from thegas streamline) One would also expect that maximum contrac-tion and signi cant particle impact loss would both occur at Staround 1 For the case of a cylindrical lens the relationships canbe expressed as

acutec D f

sup3ID

OD

L

ID Re0 St

acute [10]

acutet D g

sup3ID

OD

L

ID Re0 St

acute [11]

where ID D ID1 D ID2

To test the validity of Equations (10)ndash(11) acutec and acutet werecalculated by changing the ow eld parameters with lens di-mensions xed or changing lens dimensions with ow eld pa-rameters xed Re0 and the dimensionless lens geometry(IDOD) were kept constant in either case Operationally fora lens of xed geometry acutec and acutet were calculated by specify-ing a number of different upstream pressures For each upstreampressure a trial-and-error calculation was then undertaken to de-termine that value of the downstream pressure that would givethe same gas mass ow rate as had been found in the base caseThis is equivalent to keeping the upstream Reynolds numberconstant Results of this calculation for a thin plate lens areplotted versus particle diameter in Figure 6 where upstreamgas pressures were 140ndash340 Pa whereas lens geometry and gasmass ow rate are unchanged (ID D 5 mm OD D 10 mmL D 05 mm and Q D 60 sccmin Re0 D 125) One cansee that each curve in Figure 6 exhibits trends that are sim-ilar to those in Figure 5 In addition it is observed that themaximal collimation particle size is smaller for lower upstream

Figure 6 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle diameter for a single thin ori celens with gas upstream pressure varying from 140 Pa to 340 PaRe0 D 125 ID D 5 mm OD D 10 mm L D 05 mm Q D60 sccmin

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 621

of 001 iquest can be written as

iquest Dfrac12p D p

2Cs

18sup1[5]

where frac12p and D p are particle density and diameter respectivelyCs is a correction coef cient to Stokesrsquo law which is importantif the particle diameter is not much larger than the gas mean freepath It can be expressed as

Cs D 1 C Knp[A C Q exp(iexclB=Knp )] [6]

where Knp is the particle Knudsen number de ned as the ratioof the gas mean free path to particle diameter and A Q and Bare 121 041 and 089 respectively (Allen and Raabe 1982)Typical values of Kn in this work were 22 to 4400 so particledrag is largely associated with the free molecular ow regimefor which Cs AKnpParticle motion is usually characterized in terms of the Stokesnumber which is de ned as

St Diquest Vc

L c [7]

where Vc and L c are characteristic velocity and length scalesrespectively

Equations (5)ndash(7) together with the well-known expressionfor mean free path yields

St D

8gtgtlt

gtgt

kT

18sup1p

2frac34 P

Vc

Lcfrac12p Dp free molecule regime

1

18sup1

Vc

Lcfrac12p D2

p continuum regime

[8]

where frac34 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 (frac34 sup1 and P) gas velocity lens char-acteristic dimension and particle properties (frac12pDp ) Thereforefor a given lens system operating under constant ow conditionsin the free molecule regime the Stokes number depends on theproduct frac12pDp In the continuum regime however the functionaldependence of particle Stokes number on the particle propertiesgoes as frac12p D p

2 Note that particle diameters in this paper are pre-sented as particle aerodynamic diameter (frac12pDp with frac12p equalto one gcm3) unless speci ed

RESULTS AND DISCUSSION

Particle Motion through a Single LensNumerical analysis of a single lens was undertaken in order

to develop an understanding of effects of lens geometry and ow conditions on beam contraction This understanding canprovide general guidance for the design of a lens system For all

dimensional 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 ID1 D ID2 D 5 mm thicknessL D 05 mm and OD D 10 mm In the single lens analy-sis the upstream air pressure is 280 Pa and the ow rate Qis 60 secmin These parameters result in Re0 D 125 whereRe0 is based on the upstream quantities evaluated at X D 0The 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 the particle trajectories now exhibit a strong radialcontraction effect The bottom plot (Figure 3c) is for 10000 nmdiameter particles This plot shows that these particles have somuch inertia that their trajectories are hardly in uenced 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 ef ciency In the present analysis of a single lens particletransmission ef ciency acutet is de ned as the particle downstream ow rate divided by the upstream ow rate It is assumed thatparticles are uniformly dispersed in the air upstream of the lens

Figure 3 Particle trajectories through a single thin cylindricallens (D p D 5500 1000 nm) L D 05 mm ID1 D ID2 D 5 mmOD D 10 mm Q D 60 sccmin Re0 D 125 and Pup D 280 Pa

622 X ZHANG ET AL

Figure 4 Particle downstream radial coordinate Rpo versus upstream radial coordinate Rpi for Dp D 5 500 and 10000 nmOther parameters are the same as in Figure 3

and that any particle that impacts on the ori ce plate is lost fromthe ow For a parabolic upstream velocity pro le and purelyballistic particles the transmission ef ciency is given by

acutet b D 2

sup3ID

OD

acute2

1 iexcl12

sup3ID

OD

acute2

[9]

This provides a lower bound for the actual transmissionef ciency

In Figure 4 the particle trajectory results from Figure 3 areplotted as downstream radial-coordinate Rpo versus upstreamradial-coordinate Rpi The results for Dp D 5 nm form a straightand nearly diagonal line which con rms the lack of any signif-icant radial contraction The line for D p D 500 nm is almoststraight with a slope of about 05 This indicates that the con-traction ratio acutec is a constant equal to 05 A roughly linearrelationship between Rpo and Rpi was also observed for D p D10000 nm but with a larger slope due to higher particle inertiaThe results in Figure 4 suggest that the beam contraction ratio isroughly constant It is therefore appropriate to use the contrac-tion ratio acutec D RpoRpi at a certain Rpi to quantify the lens colli-mation effect Throughout the present work acutec at Rpi D 15 mmor RpiOD D 015 is used to represent overall lens performanceThis is similar to the near axis acutec used by Liu et al (1995a b)

Figure 5 shows results for acutec versus particle diameter for atypical lens (ID D 5 mm OD D 10 mm L D 05 mm) oper-ating at a typical condition (Pup D 280 Pa Q D 60 secminRe0 D 125) but at different grid densities One can see that eachcurve in Figure 5a starts from acutec close to unity because particleinertia is too small to produce a contraction and then reaches aminimum at a certain diameter which is referred to as the maxi-

mal collimation diameter Finally acutec increases for particles thatare too heavy to respond to the gas contractionexpansion Thissuggests that for a given lens upstream pressure particles of di-ameters around the maximal collimation diameter are ef cientlydisplaced Figure 5a demonstrates that double or half of the nor-mal grid density (R-11 gridmm X-7 gridmm squares) doesnot signi cantly alter the results (maximum deviation is about1) but using one quarter of the normal grid density produces anoticeable deviation (6) from the calculation with the normalgrid density

Figure 5b provides corresponding results for the transmissionef ciency (acutet ) versus particle diameter The particle transmis-sion ef ciency starts at unity because there is no impact lossfor small particles At a characteristic diameter at which par-ticles have appreciable displacement in the top plot there is asharp drop in the transmission ef ciency due to impact loss Fi-nally for suf ciently large particles the impact loss is simplygeometrical blocking ie particles at Rpi gt ID2 are blockedby the lens whereas all others are transmitted as described byEquation (9) Similar to the results for acutec plotted in Figure 5aFigure 5b demonstrates that double or half of the normal griddensity (R-11 gridmm X-7 gridmm) does not signi cantly al-ter the results for acutet (maximum deviation is about 25) but onequarter of the normal grid density results in clear deviation (6)from the calculation with the normal grid density It should benoted that the accuracy of acutet is also in uenced by the resolutionin Rpi (currently 01 mm) for test particles For instance for thecircles and squares at 104 nm in Figure 5b it was found that testparticles injected Rip D 25 mm passed through the lens but testparticles at Rip D 26 mm impacted acutet is calculated as 0438based on Rpi D 25 mm as the maximum radius for particles

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 623

Figure 5 Particle beam contraction ratio (a) and transmissionef ciency(b)versus particle diameter for a single thin cylindricallens calculated with different grid densities Parameters are thesame as in Figure 3

that pass through the lens The triangles are the cases for whichRpi D 26 mm particles pass through the lens but Rpi D 27 mmparticles did not and this lead to acutet D 0468 This introduces amaximum uncertainty of 6 and no effort to increase the Rpi

resolution has been attemptedFor the low speed expansion of a dilute gas-particle mixture

through a lens as shown in Figure 3 the gas exhibits little com-pressibility so the contraction ratio (acutec ) and the transmissionef ciency (acutet ) must be functions only of lens geometry gas up-stream Re0 (which describes the gas ow eld) and particle St(which describes the departure of the particle trajectory from thegas streamline) One would also expect that maximum contrac-tion and signi cant particle impact loss would both occur at Staround 1 For the case of a cylindrical lens the relationships canbe expressed as

acutec D f

sup3ID

OD

L

ID Re0 St

acute [10]

acutet D g

sup3ID

OD

L

ID Re0 St

acute [11]

where ID D ID1 D ID2

To test the validity of Equations (10)ndash(11) acutec and acutet werecalculated by changing the ow eld parameters with lens di-mensions xed or changing lens dimensions with ow eld pa-rameters xed Re0 and the dimensionless lens geometry(IDOD) were kept constant in either case Operationally fora lens of xed geometry acutec and acutet were calculated by specify-ing a number of different upstream pressures For each upstreampressure a trial-and-error calculation was then undertaken to de-termine that value of the downstream pressure that would givethe same gas mass ow rate as had been found in the base caseThis is equivalent to keeping the upstream Reynolds numberconstant Results of this calculation for a thin plate lens areplotted versus particle diameter in Figure 6 where upstreamgas pressures were 140ndash340 Pa whereas lens geometry and gasmass ow rate are unchanged (ID D 5 mm OD D 10 mmL D 05 mm and Q D 60 sccmin Re0 D 125) One cansee that each curve in Figure 6 exhibits trends that are sim-ilar to those in Figure 5 In addition it is observed that themaximal collimation particle size is smaller for lower upstream

Figure 6 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle diameter for a single thin ori celens with gas upstream pressure varying from 140 Pa to 340 PaRe0 D 125 ID D 5 mm OD D 10 mm L D 05 mm Q D60 sccmin

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

622 X ZHANG ET AL

Figure 4 Particle downstream radial coordinate Rpo versus upstream radial coordinate Rpi for Dp D 5 500 and 10000 nmOther parameters are the same as in Figure 3

and that any particle that impacts on the ori ce plate is lost fromthe ow For a parabolic upstream velocity pro le and purelyballistic particles the transmission ef ciency is given by

acutet b D 2

sup3ID

OD

acute2

1 iexcl12

sup3ID

OD

acute2

[9]

This provides a lower bound for the actual transmissionef ciency

In Figure 4 the particle trajectory results from Figure 3 areplotted as downstream radial-coordinate Rpo versus upstreamradial-coordinate Rpi The results for Dp D 5 nm form a straightand nearly diagonal line which con rms the lack of any signif-icant radial contraction The line for D p D 500 nm is almoststraight with a slope of about 05 This indicates that the con-traction ratio acutec is a constant equal to 05 A roughly linearrelationship between Rpo and Rpi was also observed for D p D10000 nm but with a larger slope due to higher particle inertiaThe results in Figure 4 suggest that the beam contraction ratio isroughly constant It is therefore appropriate to use the contrac-tion ratio acutec D RpoRpi at a certain Rpi to quantify the lens colli-mation effect Throughout the present work acutec at Rpi D 15 mmor RpiOD D 015 is used to represent overall lens performanceThis is similar to the near axis acutec used by Liu et al (1995a b)

Figure 5 shows results for acutec versus particle diameter for atypical lens (ID D 5 mm OD D 10 mm L D 05 mm) oper-ating at a typical condition (Pup D 280 Pa Q D 60 secminRe0 D 125) but at different grid densities One can see that eachcurve in Figure 5a starts from acutec close to unity because particleinertia is too small to produce a contraction and then reaches aminimum at a certain diameter which is referred to as the maxi-

mal collimation diameter Finally acutec increases for particles thatare too heavy to respond to the gas contractionexpansion Thissuggests that for a given lens upstream pressure particles of di-ameters around the maximal collimation diameter are ef cientlydisplaced Figure 5a demonstrates that double or half of the nor-mal grid density (R-11 gridmm X-7 gridmm squares) doesnot signi cantly alter the results (maximum deviation is about1) but using one quarter of the normal grid density produces anoticeable deviation (6) from the calculation with the normalgrid density

Figure 5b provides corresponding results for the transmissionef ciency (acutet ) versus particle diameter The particle transmis-sion ef ciency starts at unity because there is no impact lossfor small particles At a characteristic diameter at which par-ticles have appreciable displacement in the top plot there is asharp drop in the transmission ef ciency due to impact loss Fi-nally for suf ciently large particles the impact loss is simplygeometrical blocking ie particles at Rpi gt ID2 are blockedby the lens whereas all others are transmitted as described byEquation (9) Similar to the results for acutec plotted in Figure 5aFigure 5b demonstrates that double or half of the normal griddensity (R-11 gridmm X-7 gridmm) does not signi cantly al-ter the results for acutet (maximum deviation is about 25) but onequarter of the normal grid density results in clear deviation (6)from the calculation with the normal grid density It should benoted that the accuracy of acutet is also in uenced by the resolutionin Rpi (currently 01 mm) for test particles For instance for thecircles and squares at 104 nm in Figure 5b it was found that testparticles injected Rip D 25 mm passed through the lens but testparticles at Rip D 26 mm impacted acutet is calculated as 0438based on Rpi D 25 mm as the maximum radius for particles

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 623

Figure 5 Particle beam contraction ratio (a) and transmissionef ciency(b)versus particle diameter for a single thin cylindricallens calculated with different grid densities Parameters are thesame as in Figure 3

that pass through the lens The triangles are the cases for whichRpi D 26 mm particles pass through the lens but Rpi D 27 mmparticles did not and this lead to acutet D 0468 This introduces amaximum uncertainty of 6 and no effort to increase the Rpi

resolution has been attemptedFor the low speed expansion of a dilute gas-particle mixture

through a lens as shown in Figure 3 the gas exhibits little com-pressibility so the contraction ratio (acutec ) and the transmissionef ciency (acutet ) must be functions only of lens geometry gas up-stream Re0 (which describes the gas ow eld) and particle St(which describes the departure of the particle trajectory from thegas streamline) One would also expect that maximum contrac-tion and signi cant particle impact loss would both occur at Staround 1 For the case of a cylindrical lens the relationships canbe expressed as

acutec D f

sup3ID

OD

L

ID Re0 St

acute [10]

acutet D g

sup3ID

OD

L

ID Re0 St

acute [11]

where ID D ID1 D ID2

To test the validity of Equations (10)ndash(11) acutec and acutet werecalculated by changing the ow eld parameters with lens di-mensions xed or changing lens dimensions with ow eld pa-rameters xed Re0 and the dimensionless lens geometry(IDOD) were kept constant in either case Operationally fora lens of xed geometry acutec and acutet were calculated by specify-ing a number of different upstream pressures For each upstreampressure a trial-and-error calculation was then undertaken to de-termine that value of the downstream pressure that would givethe same gas mass ow rate as had been found in the base caseThis is equivalent to keeping the upstream Reynolds numberconstant Results of this calculation for a thin plate lens areplotted versus particle diameter in Figure 6 where upstreamgas pressures were 140ndash340 Pa whereas lens geometry and gasmass ow rate are unchanged (ID D 5 mm OD D 10 mmL D 05 mm and Q D 60 sccmin Re0 D 125) One cansee that each curve in Figure 6 exhibits trends that are sim-ilar to those in Figure 5 In addition it is observed that themaximal collimation particle size is smaller for lower upstream

Figure 6 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle diameter for a single thin ori celens with gas upstream pressure varying from 140 Pa to 340 PaRe0 D 125 ID D 5 mm OD D 10 mm L D 05 mm Q D60 sccmin

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 623

Figure 5 Particle beam contraction ratio (a) and transmissionef ciency(b)versus particle diameter for a single thin cylindricallens calculated with different grid densities Parameters are thesame as in Figure 3

that pass through the lens The triangles are the cases for whichRpi D 26 mm particles pass through the lens but Rpi D 27 mmparticles did not and this lead to acutet D 0468 This introduces amaximum uncertainty of 6 and no effort to increase the Rpi

resolution has been attemptedFor the low speed expansion of a dilute gas-particle mixture

through a lens as shown in Figure 3 the gas exhibits little com-pressibility so the contraction ratio (acutec ) and the transmissionef ciency (acutet ) must be functions only of lens geometry gas up-stream Re0 (which describes the gas ow eld) and particle St(which describes the departure of the particle trajectory from thegas streamline) One would also expect that maximum contrac-tion and signi cant particle impact loss would both occur at Staround 1 For the case of a cylindrical lens the relationships canbe expressed as

acutec D f

sup3ID

OD

L

ID Re0 St

acute [10]

acutet D g

sup3ID

OD

L

ID Re0 St

acute [11]

where ID D ID1 D ID2

To test the validity of Equations (10)ndash(11) acutec and acutet werecalculated by changing the ow eld parameters with lens di-mensions xed or changing lens dimensions with ow eld pa-rameters xed Re0 and the dimensionless lens geometry(IDOD) were kept constant in either case Operationally fora lens of xed geometry acutec and acutet were calculated by specify-ing a number of different upstream pressures For each upstreampressure a trial-and-error calculation was then undertaken to de-termine that value of the downstream pressure that would givethe same gas mass ow rate as had been found in the base caseThis is equivalent to keeping the upstream Reynolds numberconstant Results of this calculation for a thin plate lens areplotted versus particle diameter in Figure 6 where upstreamgas pressures were 140ndash340 Pa whereas lens geometry and gasmass ow rate are unchanged (ID D 5 mm OD D 10 mmL D 05 mm and Q D 60 sccmin Re0 D 125) One cansee that each curve in Figure 6 exhibits trends that are sim-ilar to those in Figure 5 In addition it is observed that themaximal collimation particle size is smaller for lower upstream

Figure 6 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle diameter for a single thin ori celens with gas upstream pressure varying from 140 Pa to 340 PaRe0 D 125 ID D 5 mm OD D 10 mm L D 05 mm Q D60 sccmin

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

624 X ZHANG ET AL

Figure 7 As in Figure 6 but plotted versus particle Stokesnumber The gure shows that both the maximum beam con-traction and signi cant particle impact loss occur at St raquoD 1

pressures and the impact loss starts at a smaller diameter at lowerpressures

The data in Figure 6 are replotted in Figure 7 versus Stokesnumber St which is de ned in Equation (7) Here L c is taken asID1 and Vc is set equal to V0 Figure 7 shows that all the curvesin Figure 6 collapse onto 2 nearly universal curves for acutec and acutet On the other hand deviations from universality are larger thanthose due to grid density effects and Rpi resolution (5 and 14versus 1 and 6 for acutec and acutet respectively) The deviations inFigure 7 are probably due mainly to differences in the pressuredrop across the lens for different upstream pressures (14 forPup D 140 Pa and 2 for Pup D 340 Pa) As a result particlecharacteristic time iquest depends not only on the upstream pres-sure but also on pressure drop Another source of uncertainty isa lack of convergence (sect2) to a value Q of 60 sccmin

In Figure 7a it is shown that the minimum acutec occurs at Staround 1 and in Figure 7b the onset of signi cant impact lossesalso occurs at St raquoD 1 As expected the results depend only onSt and Re0 A practical lens design seeks maximum particlecontraction and minimum impact loss Because of signi cantimpact loss at St raquoD 1 the results suggest that it is not appro-

priate to operate a lens at St raquoD 1 unless the particles have beenpreviously collimated Instead a reasonable operating conditionis St raquoD 02 where there is considerable trajectory contraction butno impact loss Figure 6 simply re ects the fact that the Stokesnumber in the free molecule regime (Equation (8)) is roughlyproportional to 1P2 when frac120V is held constant In additionsince the optimum particle diameter corresponds to St raquoD 02it is now obvious why aerodynamic lenses usually operate at apressure of about 300 Pa for collimating particles with diametersof about 1 micron

Equation (10) was also tested by changing the OD from 5to 30 mm and changing the ID such that IDOD D constant D04 In this calculation the average upstream gas velocity V0was chosen in such a way that Re0 D constant D 125 asshown in Figure 8 In Figure 8a the contraction ratio is plot-ted versus diameter The plot shows that the maximal colli-mation diameter is smaller if the device is smaller The con-traction ratio was plotted versus St in Figure 8b where the4 curves are seen to collapse onto 1 universal curve with mi-nor deviations This is another con rmation of Equation (10)

Figure 8 Particle beam contraction ratio versus particle dia-meter (a) and versus particle Stokes number (b) for a single thinori ce lens with ID D 2ndash12 mm IDOD D 04 Re0 D 125Pup D 280 Pa L D 05 mm Re0 D 125 Q D 60 sccmin

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 625

ie the particle contraction ratio is a function only of the lensdimensionless geometry IDOD the Reynolds number Re0and the Stokes number St In addition a comparison ofFigures 7 and 8 shows that the parameter IDOD is extremelyimportant

Near-axis contraction ratios and transmission ef ciencies atvarious upstream Re0 for a single lens are plotted versus St inFigure 9 for the case of a xed thin cylindrical lens of OD D10 mm ID1 D ID2 D 4 mm and L D 05 mm Gas downstreampressures corresponding to each Re0 are also included in the g-ure The gas upstream pressure is xed at 280 Pa As in Figure 7it is observed that the minimum values of acutec occur at St around1 signi cant impact losses also occur at St raquoD 1 It is observedin the top plot that higher Re0 led to a greater contraction andeventually acutec goes negative in the regime around the maximalcollimation diameter which means that particles moved acrossthe center line Similarly in the bottom plot there appears tobe a larger impact loss at higher Re0 An examination of thestreamlines for these same Reynolds numbers showed that theupstream contraction of the streamlines was more gradual for

Figure 9 Particle beam contraction ratio (a) and transmissionef ciency (b) versus particle Stokes number for Re0 D 72ndash517Single thin cylindrical lens with L D 02 ID D 4 mm OD D10 mm Pup D 280 Pa Q D 60 sccmin

Figure 10 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle Stokes number for IDOD D02ndash06 Single thin cylindrical lens L D 05 mm OD D 10 mmPup D 280 Pa Re0 D 125 Q D 60 sccmin

the lower values of Re0 and more abrupt for the higher valuesThe associated changes in acutec and acutet are therefore not surprising

Figure 10 shows the effect of the IDOD ratio on contractionratio and impact loss Here lens geometry and gas upstreampressure are the same as in Figures 8 and 9 except that theID was varied from 2 to 6 mm The gas ow rate was xed at60 secmin corresponding to Re0 D 125 The contraction ratioand the transmission ef ciency show trends similar to those inFigures 7ndash8 ie maximum contraction and signi cant impactloss occur at St raquoD 1 In addition the gure shows that decreasingthe ID strongly increases the extent of beam contraction and acutec

nearly reaches iexcl4 for IDOD D 02 It also can be seen thatthere is more impact loss for a smaller IDOD ratio due to thesmaller open area Because smaller values of IDOD producea more abrupt contraction of the streamlines these results arequalitatively consistent with those in Figure 9

The in uence of the lens thickness on the particle contrac-tion ratio is shown in Figure 11 Lens OD and ID are 10 and4 mm respectively and upstream pressure and Re0 are the sameas in Figure 10 The lens thickness was increased from 05 to

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

626 X ZHANG ET AL

Figure 11 Particle beam contraction ratio versus particle diameter (lower axis) and Stokes number (upper axis) for L D 05ndash

40 mm with Re0 D 125 and Q D 60 sccmin

40 mm As might be expected the gas downstream pressureswhich are shown in the gure decrease with increasing L Gen-erally it is observed that larger values of L result in strongerbeam contraction The reason for this is that a longer lens cre-ates a larger beam contraction because the effect of downstreamgas expansion is correspondingly delayed The phenomenonis shown in Figure 12 where particle trajectories for 25 sup1m

Figure 12 Particle trajectories for 25 sup1m diameter particles in lenses of LID D 0125 125 and 25

diameter particles in lenses of LID D 0125 125 and 25 areplotted

Conical lenses were also studied with the expectation thatthey would show better transmission ef ciencies than werefound with cylindrical lenses In these calculations the lensdownstream diameter ID2 was xed at 4 mm whereas the up-stream diameter ID1 was changed from 4 to 10 mm The lens

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 627

Figure 13 Particle beam contraction ratio (a) and transmis-sion ef ciency (b) versus particle diameter (lower axis) andStokes number (upper axis) for a single conical lens with frontopening diameter ID1 D 4ndash10 mm ID2 D 4 mm L D 10 mmRe0 D 125 Q D 60 sccmin

thickness was xed at 10 mm The results in Figure 13 showthat the contraction ratio is very sensitive to small departuresfrom the cylindrical case but that it soon becomes independentof the upstream diameter The transmission curves indicate thata conical lens is only modestly superior to a cylindrical lensOne can see from the gure that by implementing a conicallens the diameter range for 100 transmission can be extendedby a factor of 2

The results of single lens calculations indicate that the extentof contraction and impact loss are controlled by dimensionlessparameters IDOD LID Re0 and St Both the maximal colli-mation diameter and the conditions at which signi cant impactlosses begin can be characterized by St raquoD 1

Liu et alrsquos (1995a b) near-axis calculation also showed thatacutec decreases with increasing Re0 and increases with increasingIDOD however impact losses were not analyzed Moreovertheir results showed only a monotonic decay of acutec versus Stpossibly because their calculation was limited to St lt 3

Particle Motion in a Nozzle ExpansionAs discussed above a single-stage expansion to vacuum

through a conical nozzle has been used extensively to generateparticle beams previous studies have shown that both nozzleshape and upstream pressure in uence beam divergence Theconcept of an aerodynamic lens system was proposed by Liuet al (1995a b) as a technique to generate a collimated parti-cle beam that could then be expanded to vacuum via a super-sonic nozzle Liu et al (1995a b) also concluded that the nalnozzle geometry plays an important part in determining beamdivergence Obviously nozzle geometry also controls particleterminal velocity which is size dependent This is important ifparticle size is to be determined by measurement of the terminalvelocity For these reasons we have also studied ow throughan isolated nozzle

As shown in Figure 14 a 2 stage nozzle that was proposedby Liu et al (1995a b) has been examined In the nozzle thecross section available for gas ow is reduced in 2 stages rstfrom the tube diameter OD to dt and after a distance L to thethroat diameter dn (thickness of the throat is negligible) Thepressure ratio across the nozzle is large as the pressure dropsfrom perhaps 150 Pa to 01 Pa After the throat the gas under-goes free expansion The quality of the resulting particle beamcan be characterized by the beam divergence angle reg which isde ned as the angle between the nal particle trajectory and thejet axis The angle reg is expected to be a function of gas ow pa-rameters (upstream Reynolds number and the ratio of the up- anddownstream pressures PupPdown ) the dimensionless geometricparameters of the nozzle and the particle Stokes number Thereason for inclusion of PupPdown here is that the gas compress-ibility plays a large role in determining the dynamics if this ratiois substantially larger than 1 In particular if the ratio is about 2(or larger) the velocity at the throat can be expected to be sonicFor a lens PupPdown is close to unity so that the compressibilityeffects are negligible (Mach number is lt01) by contrast inthe nozzle expansion PupPdown is on the order of 1000 (Machnumber can reach 2) In the calculations near-axis particles atan upstream radial location of 2RpiOD D 01 (Rpi is particle ini-tial radial location) were chosen to characterize nozzle overallperformance (as used by Liu et al (1995a b)) and nozzle down-stream pressure was chosen as 01 Pa based on a typical real sys-tem (Jayne et al 2000) Calculations were rst made for a nozzlewith dn D 3 mm dt D 6 mm L D 10 mm and OD D 10 mmNozzle mass ow rate Q was 100 secmin and the nozzle up-stream pressure that matched the ow rate was 150 Pa Theparticle divergence angle is plotted as squares (marked as 11)in Figure 14 versus particle diameter (bottom axis) or particleStokes number (top axis) The other 2 sets of data in the g-ure are the cases for which the nozzle is geometrically reduced(circles marked as 051) and magni ed (triangles marked as21) by a factor of 2 in both the axial and radial dimensionsObviously the 3 nozzles are geometrically similar but have dif-ferent absolute geometrical dimensions Also in the calculationsQOD is maintained at a constant because this is equivalent to

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

628 X ZHANG ET AL

Figure 14 Particle divergence angle versus particle diameter (lower axis) and Stokes number (upper axis) for geometricallyenlarged (21) and shrunken (051) nozzles The geometric dimensions (mm) for 11 are OD D 10 dt D 6 dn D 3 L D 10 Re0 D208 Q D 100 sccmin

Re0 D constant It has been found from the calculated gas eldsthat Mach number pro les are similar in all 3 nozzles Therefore ow in the 3 cases exhibits similarity Figure 14 shows that inall 3 cases results for particle divergence angle versus particlediameter or Stokes number lie on a single curve This is proofof the hypothesis that the nozzle dimensionless parameters (notthe absolute dimensions) determine particle beam performanceSince sonic ow is expected at the throat an appropriate veloc-ity scale Vc for use in the Stokes number is the speed of soundFor convenience we used the value of the upstream sound speedFor the Reynolds number we used the upstream velocity as theappropriate scale just as in the case of the lens We also usedthe upstream value of iquest for convenience although this may bemore dif cult to justify on physical grounds As in the lens casewe used the throat diameter as the length scale Interestinglyit is observed in Figure 14 that the divergence angle is positivefor small Dp (because smaller particles follow the gas stream)but then becomes negative for large Dp (because larger particlesmay cross the axis due to inertia) The beam is least divergentfor those very large particles (Dp gt 5 sup1m) that are too heavy torespond appreciably to the gas contraction and expansion occur-ring up- and downstream of the nozzle The gure also showsthat particles cross the axis at St cedil 1

Calculations have also been performed for a nozzle of xedgeometry butwith upstream pressures varying from 75 to 600 Paand this results in Re0 varying from 86 to 98 The Mach numberpro le is found to be essentially unchanged Figure 15 shows theparticle divergence angle versus particle diameter (Figure 15a)

and versus particle Stokes number (Figure 15b) Figure 15ashows that higher Reynolds numbers (higher upstream pressure)result in smaller beam divergence for larger particles (D p gt

1000 nm)but larger beam divergence for smaller particles Whenthe data were plotted versus St in the bottom plot (Figure 15b)the 4 curves were found to be nearly coincident This suggeststhat Reynolds number has a minor in uence on the beam diver-gence whereas Stokes number has a major in uence

The effect of nozzle geometry is shown in Figure 16 wherethe calculated particle divergence angle for dt D 3ndash10 mm ispresented in Figure 16a as a function of particle Stokes num-ber Other parameters are dn D 3 mm OD D 10 mm andPdown D 01 Pa The gas mass ow rate was 100 secminfor all cases and upstream Reynolds number Re0 equals 208 forall cases Gas upstream pressures matching Q D 100 secminare also included in the gure All the curves in the gure ex-hibit a similar trend in that the curves start with positive valuescross the axis at St D 05ndash5 and nally become less divergentThe diamonds (dt D OD D 10 mm) and the circles (dt D dn D3mm) represent the limiting cases of a single step thin cylindricalnozzle and a single step thick cylindrical nozzle The commonfeature of the 2 cases is that the cross section for ow was de-creased in 1 step (at the throat for dt D OD D 10 mm caseand at the front edge of the cylinder for the dt D dn D 3 mmcase) The upside down triangles (dt D 8 mm) represent a casesimilar to the thin cylindrical nozzle as the cross section wasdecreased mostly at the throat The other 2 lines which representcases of changing the cross section in 2 more-or-less equal steps

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 629

Figure 15 Particle divergence angle versus particle diameter(a) and Stokes number (b) with upstream Reynolds number vary-ing from 86 to 978 The geometric dimensions (mm) for thenozzle are OD D 10 dt D 6 dn D 3 L D 10

have smaller beam divergence Among them the right-side-uptriangles (dt D 6 mm dt dn D 2) provide the best overall per-formance Additional calculations showed that further increasesin the number of steps did not signi cantly enhance the beamperformance

Particle terminal velocity (normalized by upstream soundspeed) is shown in Figure 16b versus St for the same valuesof the parameter dt The results indicate that normalized par-ticle velocity is much less sensitive to dt than is the beam di-vergence It is interesting to note that the normalized particlevelocity attains a value of 1 at St raquoD 1 Smaller particles (St lt 1)are accelerated to more than the upstream sound speed whereaslarger particles (St gt 1) attain lower velocities because of theirhigh inertia This result can be used to infer a particle size bymeasuring particle velocity As shown in Equation (8) the parti-cle Stokes number is proportional to frac12p Dp in the free moleculeparticle ow regime (which is valid for all conditions of thiswork) and this product is de ned as particle aerodynamic di-ameter This de nition is different from the usual one frac12p D p

2

Figure 16 Particle divergence angle as a function of particleStokes number (a) for dt D 3ndash10 mm (b) is particle terminalvelocity (normalized by upstream sonic speed) versus Stokesnumber Other parameters are OD D 10 dt D 6 dn D 3 L D 10Q D 100 sccmin 2RpiOD D 01 Re0 D 208

which is valid for the continuum regime (Baron and Willeke1993)

From a quasindashone-dimensional semiempirical model Liuet al (1995a b) calculated particle divergence angles and ter-minal velocities for particles up to 250 nm in diameter forQ D 108 secmin for a nozzle expansion To compare theirbeam divergence data with the present analysis calculationshave been performed on a nozzle of the same geometric con g-uration (dt D 6 mm dn D 3 mm L D 10 mm OD D 10 mm)and with the same ow rate (Q D 108 secmin) as Liu et alrsquosThe results are plotted along with those of Liu et al (1995b) inFigure 17 Figure 17b shows that the terminal velocity of parti-cles on the axis as predicted by Fluent and by Liu et al (1995b)are in reasonable agreement with their experimental data How-ever Figure 17a shows that the Fluent model typically predicts alarger particle divergence angle than Liu et alrsquos calculation Thereason for agreement of all 3 sets of data of terminal velocity inFigure 17b is probably because terminal velocity of particles onthe axis is almost solely determined by gas velocity at the axis

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

630 X ZHANG ET AL

Figure 17 Particle divergence angle (a) and terminal velo-city (b) versus Dioctyl Sebacate (DOS) particle geometric di-ameter OD D 10 dt D 6 dn D 3 L D 10 Q D 108 sccmin2RpiOD D 01

In Liu et alrsquos model the gas velocity on the axis was based on awell-studied and experimentally con rmed relationship betweenthe centerline Mach number and nozzle downstream distanceOn the other hand the discrepancy in Figure 17a probably sug-gests that the simple mass conservation relationship used by Liuet al (1995b) to extract radial gas velocity has some uncertain-ties These uncertainties lead to noticeable errors in beam diver-gence prodictions even for near-axis particles (2Rpi OD D 01)

CONCLUSIONSParticle motion for ows of gas-particle suspensions through

single ori ces has been investigated numerically Both com-pressible and incompressible ows have been modeled

For an incompressible ow of a gas through an ori ce theassociated particles may either experience a net displacementtoward the axis or they may impact on the front surface of thelens and be lost from the stream depending on particle inertia Itwas found that the extent of beam contraction and particle impactloss are controlled by the geometrical parameters IDOD andLID and the dynamical parameters Re0 and St The maximum

particle contraction is achieved at St raquoD 1 and signi cant impactloss also begins at St raquoD 1 The results show that a reasonableoperating condition for the lens is St raquoD 02 at which conditionbeam contraction is close to maximum but there is no impactloss These results can provide guidance for the sizing of eachlens in a multilens system The strategy is to (a) adjust IDODand Re0 to achieve maximum value of the beam contractionand (b) choose upstream ow parameters (velocity and upstreampressure) such that St raquoD 02 for the particle diameter of interestor operate the lens at St raquoD 1 if particles are already collimated

For the nal expansion through a nozzle from the lens work-ing pressure (raquo150 Pa) to vacuum (01 Pa) it was found thatparticle beam divergence is a function of Re0 nozzle geometryand St and among these the nozzle geometry and Stokes num-ber have major impacts It was also found that a stepped nozzlegenerally helps to reduce beam divergence The calculated par-ticle terminal velocity was shown to be in good agreement withthe experimental data of Liu et al (1995b) (Figure 17b) andseparately by Jayne et al (2000)

REFERENCESAllen M D and Raabe O G (1982) Re-Evaluation of Millikanrsquos Oil Drop

Data for the Motion of Small Particle in Air J Aerosol Sci 13537ndash547Baron P A and Willeke K (1993) Gas and Particle Motion In Aerosol

Measurement edited by P A Baron and K Willeke John Wiley amp SonsNew York pp 23ndash53

Dahneke B E and Cheng Y S (1979a) Properties of Continuum SourceParticle Beam I Calculation Methods and Results J Aerosol Sci 10257ndash

274Dahneke B E and Cheng Y S (1979b) Properties of Continuum Source

Particle Beam II Beams Generated in Capillary Expansion J Aerosol Sci10363ndash368

Davis W D (1977) Continuous Mass Spectrometer of Airborne Particulatesby Use of Surface Ionization Environ Sci Technol 11587ndash592

Dahneke B E and Flachsbart H (1972) An Aerosol Beam SpectrometerJ Aerosol Sci 3345ndash349

Estes T J Vilker V L and Friendlander S K (1983) Characteristics of aCapillary Generated Particle Beam J Colloid Interface Sci 9384ndash94

Henry C (1998) Dust in the Wind Anal Chem 1462Andash465AIsrael G W and Friedlander S K (1967) High Speed Beams of Small

Particles J Colloid Interface Sci 24330ndash337Jayne J T Leard D L Zhang X Davidovits P Smith K A Kolb C E and

Worsnop D R (2000) Development of an Aerosol Mass Spectrometer forSize and Composition Analysis of Submicron Particles Aerosol Sci Technol3349ndash70

Johnston M V and Wexler A S (1995) Mass Spectrometry of IndividualAerosol Particles Anal Chem 67721Andash726A

Kantrowitz A and Grey J (1951) A High Intensity Source of the MolecularBeam Review of Scienti c Instruments 22328ndash332

Kievit O Weiss M Verheijet P J T Marijnissen J C M and Scarlett B(1996) The On-Line Chemical Analysis of Single Particles Using AerosolBeams and Time of Flight Mass Spectrometer Chem Eng Comm 15179ndash

100Liu P Ziemann P L Kittelson D B and McMurry P H (1995a) Generat-

ing Particle Beams of Controlled Dimensions and Divergence I Theory ofParticle Motion in Aerodynamic Lenses and Nozzle Expansions Aerosol SciTechnol 22293ndash313

Liu PZiemann P L Kittelson D B and McMurry P H (1995b) GeneratingParticle Beams of Controlled Dimensions and Divergence II Experimental

A NUMERICAL ANALYSIS OF AN AERODYNAMIC LENS 631

Evaluation of Particle Motion in Aerodynamic Lenses and Nozzle Expan-sions Aerosol Sci Technol 22314ndash324

Mallina R V Wexler A S and Johnston M V (1999) High Speed ParticleBeam Generation Simple Focusing Mechanisms J Aerosol Sci 30719ndash738

Mallina R V Wexler A S Rhoads K P and Johnston M V (2000)High Speed Particle Beam Generation A Dynamic Focusing Mech-anism for Selecting Ultra ne Particles Aerosol Sci Technol 3387ndash

104Murphy D M and Thomson D S (1995) Laser Ionization Mass Spectrometry

of Single Aerosol Particles Aerosol Sci Technol 22237ndash249Murphy W K and Sears G W (1964) Production of Particle Beams J Appl

Phys 851986 ndash1987Noble C A and Prather K A (1996) Real-Time Measurement of Correlated

Size and Composition Pro les of Individual Atmospheric Aerosol ParticlesEnviron Sci Technol 302667ndash2680

Rao N P Navascues J and Mora J F (1993) Aerodynamic Focusing ofParticles in Viscous Jets J Aerosol Sci 24879ndash892

Shreiner J Voigt C Mauersberger K McMurry P and Ziemann P (1998)Aerodynamic Lens System for Producing Particle Beams at StratosphericPressures Aerosol Sci Technol 2950ndash56

Shreiner J Schild U Voigt C and Mauersberger K (1999) Focusing ofAerosol into a Beam at Pressures from 10 to 150 Torr Aerosol Sci Technol31373ndash382

Thomson D S Middlebrook A M and Murphy D M (1997) Threshold forLaser Induced Ion Formation from Aerosols in a Vacuum Using Ultravioletand Vacuum Ultraviolet Laser Wavelengths Aerosol Sci Technol 26544ndash

559Tobias H J Kooima P M Dochery K S and Ziemann P J (2000) Real-

Time Chemical Analysis of Organic Aerosols Using a Thermal DesorptionParticle Beam Mass Spectrometer Aerosol Sci Technol 33170ndash190