Three-Dimensional Simulation of Brownian Motion of Nano-Particles In Aerodynamic Lenses

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Three-Dimensional Simulation of Brownian Motion ofNano-Particles In Aerodynamic LensesOmid Abouali a , Abbas Nikbakht a , Goodarz Ahmadi b & Saieedeh Saadabadi aa Department of Mechanical Engineering , Shiraz University , Shiraz, Iranb Department of Mechanical and Aeronautical Engineering , Clarkson University , Potsdam,New York, USAPublished online: 09 Feb 2009.

To cite this article: Omid Abouali , Abbas Nikbakht , Goodarz Ahmadi & Saieedeh Saadabadi (2009) Three-DimensionalSimulation of Brownian Motion of Nano-Particles In Aerodynamic Lenses, Aerosol Science and Technology, 43:3, 205-215, DOI:10.1080/02786820802587888

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Aerosol Science and Technology, 43:205–215, 2009Copyright c© American Association for Aerosol ResearchISSN: 0278-6826 print / 1521-7388 onlineDOI: 10.1080/02786820802587888

Three-Dimensional Simulation of Brownian Motionof Nano-Particles In Aerodynamic Lenses

Omid Abouali,1 Abbas Nikbakht,1 Goodarz Ahmadi,2 and Saieedeh Saadabadi1

1Department of Mechanical Engineering, Shiraz University, Shiraz, Iran2Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, New York,USA

A computer code for analyzing nano-particle motions in an aero-dynamic particle beam focusing system was developed. The codeuses an accurate three-dimensional model for the Brownian dif-fusion of nano-particles in strongly varying pressure field in theaerodynamic lens system. Lagrangian particle trajectory analysiswas performed assuming a one-way coupling model. The particleequation of motion used included drag and Brownian forces.

The prediction of the 3-D model for penetration efficiency, beamdivergence angle, beam diameter and radial cumulative fractionwere evaluated and were compared with those of the axisymmetricmodels. The simulation results showed that for particle diametersless than 30 nm in helium gas, the Brownian force could signifi-cantly affect the beam focusing and particle penetration efficiency.Some potential errors in the naı̈ve usage of axisymmetric modelwere discussed. It was shown that the earlier axisymmetric modelslead to the incorrect mean square radial displacement of Brownianparticles. The present 3-D approach, however, leads to the correctvalue of the radial mean square displacement of 4Dt.

The effect of the inlet orifice and relaxation region on the perfor-mance of the lens system was also investigated. It was shown thatthe major losses of the 4 to 10 nm particles occur in the inlet orificeand relaxation region walls. For 15 to 30 nm particles, the mainlosses occur at the inlet orifice walls. Some alterations of the shapeof the inlet orifice were examined and a new design is suggested toreduce the loss of the particles at the inlet flow control orifice.

INTRODUCTIONAerodynamic lenses are formed by combinations of prop-

erly designed axisymmetric contractions and expansions, and

Received 7 May 2008; accepted 29 October 2008.The work of OA and AN was supported by a grant from Iranian

Nanotechnology Initiative Council (INI) and the work of GA was sup-ported by a grant from US Department of Energy (NETL) and by NYS-TAR through the Center for Advanced Material Processing (CAMP) ofClarkson University.

The authors would like to thank Professor McMurry of Universityof Minnesota and Dr. Wang for providing the detail design of their inletorifice.

Address correspondence to Omid Abouali, Department of Mechan-ical Engineering, Shiraz University, Shiraz, Fars 71348-51154, Iran.E-mail: [email protected]

are used for generating focused particle beams. Particles in acritical size range passing through a sequence of contractionsdrift towards the axis and form a narrow beam. Particles largerthan the critical size range are removed from the stream by in-ertial impaction on the lens wall, while smaller particles followthe flow streamlines and are not focused. The diameters of thefocused particle beams are controlled with appropriate sizingof sequence of lenses of varying diameters, operating pressure,Brownian motion, and particle shape.

Jayne et al. (2000), Zhang et al. (2002), Kane and John-ston (2000), and Kane et al. (2001) described the use of aero-dynamic lenses for producing focused aerosol particle beamsfor online characterization of fine particles. Murphy and Sears(1986–1987) reported producing narrow particle beams by ex-panding an aerosol from atmospheric pressure to low backpres-sures through a nozzle.

Computational and experimental studies of aerodynamiclenses were first performed by Liu et al. (1995a, b). They showedhighly collimated particle beams could be produced withoutsheath air. They also investigated the effect of Brownian mo-tion and lift force on the particle beam diameter downstream ofthe nozzle. Absence of multidimensional gas velocity field re-duces the accuracy of their particle trajectory analysis especiallyaway from the axis.

Zhang et al. (2002) characterized particle beam collimation ina single aerodynamic lens and an individual nozzle. They foundthat the maximum particle displacement and particle loss occursat a particle Stokes number near unity. The performance char-acteristics of the lens and the nozzle were found to depend ontheir geometry, flow Reynolds number, and particle Stokes num-ber. Zhang et al. (2004) studied the gas–particle flows throughan integrated aerodynamic-lens–nozzle. They found that the in-let transmission efficiency (ηt) for particles in the intermediatesize range with diameters of 30–500 nm was about unity. (Theydefined the inlet transmission efficiency as the fraction of theparticles reaching the target relative to the number of particlesat the inlet upstream. Here the target was a 2 mm diameter platein a mass spectrometer, which is located 240 mm downstreamof the nozzle.) The transmission efficiency gradually reduced to

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206 O. ABOUALI ET AL.

about 40% for large particles and to almost zero for very smallparticles with dp ∼ 15 nm. Approximation in the Brownian forcecomputation was the major limitation of their work.

Wang et al. (2005) developed a numerical simulation method-ology that was able to accurately characterize the focusing per-formance of aerodynamic lens systems. They demonstrated theability of aerodynamic lenses to focus sub 30 nm spherical unitdensity particles by a new design and arrangements of lenses andusing helium as the carrier gas. Wang et al. (2006a) developedan Aerodynamic Lens Calculator Module for design of aerody-namic lens systems. The calculator module could be used to de-sign the key dimensions of a lens system including pressure lim-iting orifice, relaxation chamber, focusing elements, spacers, andthe accelerating nozzle. Wang et al. (2006b) presented an experi-mental study of nanoparticle focusing using aerodynamic lenseswith helium as carrier gas. Measurements were done for threetypes of particle materials: vacuum pump oil, sodium chloride,and proteins in the 3–30 nm diameter range. The particle trans-port efficiency through the aerodynamic lens and the beam widthdownstream of the lens system were measured. Their resultsshowed that their lens system can deliver nanoparticles in thesize range of 3–30 nm from atmosphere pressure to high vacuumwith high transmission efficiencies and narrow beam diameters.

All earlier computational modeling studies of aerodynamiclenses were restricted to the assumption of axisymmetric flowswhere the radial particle trajectories were evaluated. The aero-dynamic lenses typically include a vacuum pump attached tothe sidewall of the intermediate chamber that makes the flowthree-dimensional (non-axisymmetric) at least in the intermedi-ate chamber. When the flow is not axisymmetric, then the particletrajectories also deviate from being axially symmetric. In addi-tion, the Brownian motion is generated by three-dimensionalstochastic excitations. The analysis of trajectories of Brown-ian particles in axisymmetric flows, however, requires certaincare. Abouali and Ahmadi (2007) presented a simulation usinga three-dimensional computer model for the airflow field andparticle motion in aerodynamic lenses. Their simulation resultsshowed that the assumption of axisymmetric flow downstreamof the nozzle was a reasonable approximation when the suctionpump of the intermediate chamber was installed far from thenozzle. They, however, used a constant slip correction factorin the expression for the Brownian force, which rendered theirsimulation results for sub 10 nm particles to a qualitative descrip-tion. Wang et al. (2005) augmented FLUENTTM by a UDF (UserDefined Function) and included the variable slip correction fac-tor in their simulation of nano-particle motions in aerodynamiclenses. Their study, however, were limited to an axisymmetricanalysis of radial particle trajectories.

In this study a three-dimensional computer model for particlemotion in aerodynamic lenses was developed. The axisymmetricform of this computer model was developed before by Nikbakhtet al. (2007). For evaluating the axisymmetric flow field in theaerodynamic lens and downstream of the nozzle, FLUENTTM

(version 6) software was used. To avoid the limitation of the

commercial software, in the present study, a new computer codefor analyzing the 3-D particle trajectories in multistage aero-dynamic lenses was developed. The new code included the ap-propriate variable slip correction factor in the expression forthe drag force and the Brownian excitation. Particle trajecto-ries inside the inlet orifice, relaxation region and lenses, throughthe outlet nozzle, and downstream of the nozzle were analyzedusing the developed code. The results of 3-D model were com-pared with those of axisymmetric model and discussed. It wasshown that the earlier axisymmetric models of particles motionin the aerodynamic lenses predict higher penetration efficiency,lower particle beam radius, and lower beam divergence anglecompared with the present 3-D simulation results. (Note thatthe penetration efficiency is defined as the ratio of the particlespassing through the nozzle to the number of particles enteringthe aerodynamic lens system.) The effect of the pressure reduc-ing orifice and relaxation region on the performance of the lenssystem was also evaluated.

MODEL DESCRIPTIONThe aerodynamic lens system designed by Wang et al.

(2006b) composed of a pressure reducing orifice, a relaxationchamber, multistage focusing elements and an outlet nozzle. Theaerosol flow passes through the inlet orifice, which reduces thepressure from atmospheric pressure to the lens operating pres-sure of about 520 Pa. Particles are focused into a tight beam asthey pass through the aerodynamic lens system and are deliveredinto a vacuum chamber.

The cross sectional and perspective views of the schematic ofthe aerodynamic lens system is shown in Figure 1. Wang et al.(2005) described a computational model of the aerodynamiclens system without the inlet orifice and relaxation chamber.Here, both cases including and excluding the orifice and relax-ation chamber in the lens system are studied. The model of thelens system without the pressure-reducing orifice and relaxationregion is studied first and the results of the 3-D and the axisym-metric models are compared. The complete model of the lenssystem including the inlet orifice and relaxation chamber is alsostudied.

The diameter of the lens tube is 10 mm and adjacent focusingelements are 15 mm apart. The contraction diameters of the threeorifices that are used in the analysis are, respectively, 1.26, 1.64,and 2.33 mm. At the end of the three focusing elements, the gaspasses through a nozzle with a diameter of 2.76 mm. The outletboundary is also shown in Figure 1 and a chamber pressure of 1Paat the nozzle downstream is imposed. A pressure-reducing ori-fice (O’keefe, 7 sapphire orifice type J, aperture diameter 71 µm)limits the aerosol flow rate with helium as the carrier gas into thelens system at 88 cm3/min. The pressure drops from atmospherepressure to about 520 Pa after the pressure-reducing orifice waspredicted by the simulation, which is in agreement with the mea-surement reported by Wang et al. (2006b). It should be empha-sized that a mass flow rate boundary condition was used at theinlet of the pressure-reducing orifice in the numerical simulation.

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3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES 207

FIG. 1. (a) Cross section of the aerodynamic lens system including the inlet orifice and relaxation chamber proposed by Wang et al. (2006b) and used in presentstudy. (b) Wire-frame view of the aerodynamic lens system without the inlet orifice and the relaxation chamber used in the present study for comparisons of 3-Dand axisymmetric models. (Figure provide in color online.)

The mean free path of the gas downstream of the nozzle isof the order of the nozzle size. While using the continuum flowassumption is questionable, nevertheless, it was used as a roughapproximation. Earlier studies by Abouali and Ahmadi (2007)showed that the flow in the lens and the nozzle is axisymmet-ric, and the flow downstream of the nozzle can also be treatedapproximately as being axisymmetric.

GOVERNING EQUATIONSFor a dilute gas-particle flow in an aerodynamic lens, a one-

way interaction model is used. That is, it is assumed while gaswill carry the particles, the concentration and size of the particlesare too small to affect the gas flow. Under this condition, the gasflow field can first be evaluated and then be used for evaluation ofthe particle trajectories. The axisymmetric compressible viscouslaminar flow field was evaluated using the upwinding method.Details of the governing equations for an axisymmetric flowcondition are given in FLUENTTM Users’ Guide (2003), andhence need not be repeated here.

The Lagrangian equation of particle motion in Cartesian co-ordinate is given as

dVp

dt= FD + FB + g, [1]

dxp

dt= Vp, [2]

where Vp is the particle velocity vector, FD is the drag forceper unit mass, FB is the Brownian force per unit mass, g is theacceleration of gravity and xp is the particle position vector.Note that the flow is in laminar regime and the particles areassumed spherical. In the 3-D model, the particle equations ofmotion were solved in Cartesian coordinates and the motionof the particles in cylindrical coordinate were evaluated usingthe standard transformation, r2 = x2 + y2, θ = tan−1(y/x),z = z.

Drag ForceThe expression for the modified Stokes drag force per unit

mass including the Cunningham correction is given by:

FD = 3µCD Re

4ρpd2pCc

(V − Vp), [3]

where V is the fluid velocity vector, dp is particle diameter, ρp

is the particle density, µ is the coefficient of viscosity, Cc is the

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208 O. ABOUALI ET AL.

Cunningham correction factor given as (Hinds 1999)

Cc = 1 + 2λ

dp[1.257 + 0.4 exp(−1.1(dp/2λ))], [4]

and Re, the particle Reynolds number, is defined as

Re = |V − Vp|dp

ν, [5]

where ν is the kinematic viscosity of the gas. In Equation (3), thedrag coefficient CD, which accounts for the Reynolds numbercorrection to the Stokes drag, is given as (Hinds 1999)

CD = 24

Re(1 + 0.15Re0.687). [6]

Note that the nonlinear corrections are expected to be signif-icant only for larger particles with appreciable slip velocity. Fornano-particles with small slip velocity, Re is very small and thesecond term in Equation (6) is negligible. The exception is forthe region at the orifices where the gas-solid slip velocity maybe large. In most parts of the lenses system, however, the slipvelocity is very small.

In Equation (4) the Cunningham correction factor depends ongas mean free path. There is a significant change in the gas meanfree path in the aerodynamic lens due to the large variations in thegas pressure. Therefore, the Cunningham correction factor mustbe calculated using the correct gas mean free path along particletrajectories for an accurate simulation of the drag and Brownianforces. The mean free path of helium is given as (Hirschfelderet al. 1954)

λ(µm) = kT√2πd2

m P= 45.3T

P. [7]

In Equation (7), dm , k, P, and T, respectively, are the heliummolecule diameter (0.258 nm), the Boltzmann constant, the gaspressure (Pascal), and the gas temperature (Kelvin).

Brownian ForceWhen a small particle is suspended in a fluid, it is subjected to

the imbalanced random impacts of the gas molecules that causethe nano-particles to move on an erratic path, which is known asthe Brownian motion. A Gaussian white noise stochastic processcan model the random impacts of the molecules. White noise isa zero mean Gaussian random process with a constant powerspectrum given by (Li and Ahmadi 1993)

Snn = 2kTβ

πm p= 2β2 D

π, with D = kT

βm= kT Cc

3πµdp[8]

where k = 1.38 × 10–23 J/K is the Boltzmann constant,β = 3πµdp/Ccm p is the inverse of the particle relaxation time,

m p is the mass of the particle, subscript n denotes the white noiseexcitation, and D is the diffusion coefficient. The procedure sug-gested by Ounis and Ahmadi (1993) and Li and Ahmadi (1993)for simulating the Brownian motion was used in the computercode developed in this study.

To start the process, a time step �t is selected. (The time stepshould be much smaller than the particle relaxation time.) Thenpairs of uniform random numbers U1 and U2 (between 0 and 1)are generated and are transformed to a pair of unit variance zeromean Gaussian random numbers. This is done using the Box-Muller transformations (Box and Muller 1958; Papoulis 1965):

G1 =√

−2 ln U1 cos (2πU2) , [9]

G2 =√

−2 ln U1 sin (2πU2) . [10]

The amplitude of the Brownian force per unit mass in x, y,and z directions at each time step is then evaluated by

Fbi = G

√π Snn

�t, [11]

where Snn is given by Equation (8), and G is a unit variance,zero mean Gaussian random number given in Equations (9) and(10). The entire generated sample of the Brownian force is thenshifted by U�t, where U is a uniform random number betweenzero and one.

RESULTSUsing a computational modeling approach, the flow fields in

a class of aerodynamic lenses for the configuration designed byWang et al. (2005, 2006b) are simulated. The grid sensitivityanalysis was performed and the size of the grid was increaseduntil the solution was independent of further refinement of thegrid. Consequently, a computational grid of 70,000 elementswas used for simulating gas flow in the axisymmetric modelof the aerodynamic lens without the inlet orifice, and 120,000elements were used for complete model of lens system includingthe pressure reducing inlet orifice. The CPU time for the abovecases was 3 and 10 h, respectively, on a 2 GHz P 4 CPU.

A mass flow rate boundary condition (2.98 × 10–7 kg/s) forhelium at the inlet of the aerodynamic lens system was used. Thepressure drops from atmosphere pressure to about 520 Pa afterthe pressure-reducing orifice in the complete model of the lenssystem was predicted by the simulation, which is in agreementwith the measurement reported by Wang et al. 2006b. Fixed pres-sure boundary condition of 1 Pa was imposed at the exit bound-ary of computational domain downstream of the nozzle. Theviscosity of the helium was computed based on kinetic theory.

In the following sections, the differences for particle motionin the 3-D and axisymmetric models for the lens system withoutthe inlet orifice and the relaxation chamber are first discussed.This is followed by the analysis of the inlet orifice and relaxationregion, using the complete model of the lens system.

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3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES 209

A) Comparisons of 3-D and Earlier Axisymmetric ModelsIn this section, the computational results for the particle mo-

tions in the gas stream of the lens system without the inlet orificeare presented. Figure 2 shows sample particle trajectories in the

aerodynamic lens system for particle diameters of 2, 6, and 10nm with and without inclusion of the Brownian force. (In thesefigures for illustrative purposes only 25 particles are injected atthe inlet.) It is seen that most of the 2 nm particles are deposited

FIG. 2. Sample trajectories through the aerodynamic lens system with and without Brownian motion for 2, 6, and 10 nm particles. (Figure provided in coloronline.)

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210 O. ABOUALI ET AL.

on the walls due to Brownian diffusion. In the absence of Brown-ian diffusion all 2 nm particles pass through the nozzle. Diffusionappears to be significant for 6 and 10 nm particles, but these par-ticles stay closer to the lens axis with no particles being lost tothe lens walls. For 30 nm particles and larger (not shown here)no significant differences can be seen between the cases withand without Brownian motion. So 30 nm particles are highlyfocused near the axis and a small skimmer can be used betweenthe aerodynamic lens system and the detection chamber. Thisfigure shows that the Brownian effects reduces the effectivenessof the aerodynamic beam focusing for particles of the order of10 nm or smaller for the range of pressure condition consideredin the present study.

To check the accuracy of the Brownian simulation procedure,a series of simulations were performed and dispersion of nano-particles injected from the centerline of a constant velocity pipeand channel flows was analyzed. The simulation results for 3-Dand axisymmetric flow in a pipe and channel flow models arecompared with the exact solution of the diffusion equation forthese simple cases. Figure 3 show the dispersion of 100 nm parti-cles with a particle-to-fluid density ratio of 2000 and flow veloc-ity of 1 m/s. For evaluating the mean-square displacement of theparticles shown in Figure 3, ensembles of 5000 and 1000 sam-ples, respectively, for the 3-D and axisymmetric models wereused.

The variances of particle displacement due to Brownian dif-fusion in Cartesian coordinate direction xi as obtained from thesolution of the diffusion equation is given as

σ 2xi

= 2Dt, [12]

FIG. 3. Comparison of the simulated mean-square particles radial and verticaldisplacements as predicted by axisymmetric and 3-D models with the exactsolutions for 100 nm particles injected at the axis of a pipe with a uniformflow. (Figure provided in color online.)

where D is the particle diffusivity given by Equation (8). Thevariance of the radial displacement of the particles that are emit-ted from a source at r = 0 can be found form the solution of thediffusion equation in cylindrical coordinates given as

σ 2r = 4Dt. [13]

As noted before, the mean square radial displacement of theBrownian particles is twice that of one-dimensional Cartesiandirection This result also follows from the fact that r2 = x2+ y2,the mean-square displacement in the radial direction is given asr2 = x2 + y2 = 2x2 = 4Dt , where the isotropy of the Brownianmovement is assumed. Abouali and Ahmadi (2007) providedadditional discussion of these results.

Figure 3 shows that the time evolution of the mean-squareradial displacements of the Brownian particles predicted by thepresent 3-D Lagrangian model is in good agreement with theexact solution given by Equation (13). Similarly the mean-squaredisplacement in the y-direction obtained by the two-dimensionalLagrangian model for two-dimensional Cartesian coordinates isin close agreement with the expression given by Equation (12).These results show that the present procedure for simulatingBrownian motions in y and R directions are quite accurate.

To provide further validation of the developed 3-D computermodel and comparison with the earlier axisymmetric models, aseries of simulation for particle transport in a pipe under lam-inar flow conditions were performed. A 10 mm diameter tubewith a length of 20 mm at a pressure of 500 Pa with averagevelocity of 5 m/s was analyzed. The simulation results of earlieraxisymmetric models, as well as the 3-D model for penetrationefficiencies are compared with those predicted by the empiricalcorrelation of Gormley–Kennedy (Gormley and Kennedy 1949)in Figure 4. This figure shows that the 3-D model predictionsare in good agreement with the experimental data. The earlieraxisymmetric model, however, overestimates the penetration for

FIG. 4. Comparison of experimental data with axisymmetric and 3-D modelsfor penetration of particles. (Figure provided in color online.)

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3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES 211

FIG. 5. Comparison of beam divergence angles as predicted by axisymmetric,Wang et al. (2005) and 3-D models. (Figure provided in color online.)

the ranges of diffusion dominant nano-particles. This is as ex-pected because the earlier axisymmetric models underestimatethe diffusion of particles in radial direction, because their pre-dicted mean-square radial displacements converge to 2Dt andunderestimate the correct value by a factor of two.

The main problem of these earlier axisymmetric models forparticle diffusion analysis in aerodynamic lenses and pipe flowsis that they use a 2-D Cartesian equation for particle motion inradial direction. Earlier, Hagwood et al. (1999) used Ito’s cal-culus for deriving an equation for stochastic particle motion inradial direction. Their analysis leads to the addition of a D/rterm to the right-hand side of the governing equation. How-ever, since D/r is singular at the axis, the computational mod-eling of their equation encounters serious difficulty for appli-cation to the configurations that dispersion near the axis is ofinterest.

Particle beam divergence angles as defined by Wang et al.(2005) were evaluated. Accordingly, the particle beam diameteris twice the radius that contains 90% of particles. For evaluatingthe beam divergence angle, the beam radius at two planes areevaluated and used. Here the plane were selected the same asWang et al. (2005) at axial distances of 75.3 mm and 85.3 mmfrom the inlet, which correspond to 1.8 and 5.4 nozzle diametersdownstream of the nozzle exit. Then the difference of these twobeam radii was divided by the distance between the two planes(here 10 mm) for evaluating the beam divergence angle. Figure5 compares the simulation results for the divergence angle of the3-D model with the predictions of earlier axisymmetric modelsand those of Wang et al. (2005). The theoretical Brownian limitderived by Liu et al. (1995a) is also shown in this figure. Theearlier axisymmetric model that underestimates radial directionforces leads to smaller values of the beam divergence angles.The prediction of Wang et al. (2005), is even smaller than to theearlier axisymmetric model, but it is close to the Brownian limitsuggested by Liu et al. (1995a).

FIG. 6. Comparison of predicted particle beam diameters by axisymmetricand 3-D models with the experimental data at 71 mm downstream of the nozzle.(Figure provided in color online.)

Figure 6 compares the present 3-D and earlier axisymmet-ric simulation results with the experimental data of Wang etal. (2006b) for particle beam diameter at 71 mm axial loca-tion downstream of the nozzle. The simulation result of Wanget al. (2005) is also shown in this figure for comparison. Allsimulations seem to capture the trend of the experimental databut somewhat underestimate the magnitude of the experimentalbeam diameters. The present model seems to show slightly bet-ter agreement with the experimental data when compared withthe prediction of Wang et al. (2005).

Figure 7 compares the radial distribution of the cumulativefraction of the particles at injection and at the nozzle exit planesas predicted by the present 3-D with the results of Wang et al.(2005). The cumulative particle fraction was normalized to thetotal number of injected particles (20000 for 3-D model). In ad-dition, as suggested by Wang et al. (2005), the particle radiallocation was normalized using the inner radius of spacers (5mm) for the inlet curve and the nozzle radius (1.38 mm) forthe outlet curves. It is seen that the cumulative fraction of theparticles injected at the inlet for all models is proportional to[2(r/R)2-(r/R)4], which is required for a uniform distribution ofthe particles in a fully developed laminar flow. As noted beforethe particle beam width is defined as the beam diameter that con-tains 90% of total particles passing through a section so particlebeam diameter can be evaluated from Figure 7. From this figure,it is seen that Wang et al. (2005) approach predicts a higher cu-mulative fraction for 10 nm particles when compared with thepresent 3-D model.

B) Complete Model of Lens System Including the Inlet OrificeIn this section the results of a series of simulations for the

complete model of the lens system is presented. Here the inletorifice and the relaxation chamber are included in the computermodel. The inlet orifice reduces the inlet atmospheric pressure to

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212 O. ABOUALI ET AL.

FIG. 7. Cumulative fraction of 2, 6, 10, and 30 nm particles at the inlet of the aerodynamic lens system and at the nozzle exit. (Figure provided in color online.)

a low pressure (520 Pa) in the relaxation region. Figure 8 showsthe velocity distribution along the axis of the lens system. It isseen that the helium gas flow reaches to a hypersonic velocityof 1700 m/s downstream of the pressure-reducing orifice. Thisvelocity is even greater than that at downstream of the nozzle inthe lens system.

Figure 9 compares the predicted penetration efficiencies forthe lens system with and without the inlet orifice and the relax-ation chamber with the experimental data of Wang et al. (2006b).It is seen that when the inlet orifice and the relaxation chamberare not included in the computational model the penetration ef-ficiency is overestimated. The complete model that includes thepressure-reducing inlet orifice and the relaxation chamber, how-ever, predicts much lower penetration efficiency that is in excel-lent agreement with the experimental data. These results confirmthat the pressure-reducing orifice and the relaxation region playan important role in penetration efficiency of nano-particles inlens system.

Figure 10 shows the percent particle losses in the orifice andrelaxation regions. Summation of the particle losses in the inlet

orifice and relaxation regions is about 60% for 4 nm particles anddrops to about 20% for 10 nm particles. This shows that mostof the particle losses occur in the inlet orifice and the relaxationregions. Comparison of the summation of the losses in the inletorifice and relaxation regions with the total loss in Figure 10shows that most particles smaller than 10 nm are captured inthe inlet orifice and relaxation regions. Particle larger than 10nm, however, are mainly captured in the pressure reducing inletregion.

Figure 11 compares the predicted variations of the beam di-ameters with the experimental data of Wang et al. (2006). Thisfigure shows that the inclusion of the inlet orifice and the re-laxation chamber does not affect the model predictions for theparticle beam diameter.

C) Comparison of Performance of Different Pressure-ReducingOrifices

The presented results showed that most of the particle lossesoccur in the inlet orifice and the relaxation chamber. Therefore, aseries of simulation for different inlet orifices and the relaxation

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3-D SIMULATION OF BROWNIAN MOTION IN AERODYNAMIC LENSES 213

FIG. 8. Velocity distribution on the axis of the complete lens system includingthe pressure-reducing orifice. (Figure provided in color online.)

FIG. 9. Comparison of the penetration efficiency for the lens system with andwithout inlet orifice and relaxation chamber. (Figure provided in color online.)

FIG. 10. Comparison of the percent losses in the inlet orifice and the relaxationregion. (Figure provided in color online.)

FIG. 11. Comparison of predicted particle beam diameters for the lens sys-tem with and without the inlet orifice and the relaxation chamber at 71 mmdownstream of the inlet. (Figure provided in color online.)

chambers was performed. Several different designs of the inletorifice as shown in Figure 12 were studied. For sake of com-parison, the minimum size of the throat diameter for all fiveorifices are kept fixed at 71 µm. The corresponding computed

FIG. 12. Five different shapes of the studied pressure-reducing orifices(Figure provided in color online.)

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214 O. ABOUALI ET AL.

FIG. 13. Comparison of the transmission efficiency for the lens system withdifferent orifice configurations. (Figure provided in color online.)

transmission efficiencies were compared with that of the O’keefesapphire orifice used by Wang et al. (2006) in Figure 13. Amongfive simulated orifice configurations, the one with a conical di-verging section (4 in Figure 12) leads to the minimum loss offine particles (smaller than 10 nm). This orifice (4 in Figure12) has a better transmission efficiency compared with that ofO’keefe sapphire orifice for sub-10 nm particles. For particlesin a range of 10–30 nm, the orifice with a conical convergingthroat (3 in Figure 12) has better transmission efficiency. Theorifices with a simple sudden expansion (2 in Figure 12) andwith a converging-diverging throat (5 in Figure 12) have lowertransmission performance compared with the original O’keefesapphire orifice.

CONCLUSIONSIn the present study, three-dimensional particle motions in

a multistage aerodynamic lens with the end nozzle and nozzledownstream were studied. Simulation results for beam diver-gence angle and particle beam diameter were presented and theresults were compared with those of the earlier axisymmetricmodels.

The Brownian simulation procedure used in the analysis forthree dimensional model was verified by comparison with theexact solutions for uniform flows. Comparison of the presentsimulation results with available models showed certain inade-quacies of the earlier axisymmetric models. In particular, someearlier axisymmetric models led to the incorrect mean-squareradial displacement of 2Dt. The present 3-D model, however,predicted the correct particle radial displacement variance of4Dt.

The effect of the inlet pressure-reducing orifice and the up-stream relaxation region on the performance of the lens systemwas also investigated. It was shown that an extremely high ve-

locity region occurs downstream of the inlet orifice. In addi-tion, most of the 4–10 nm particle losses occur in the pressure-reducing orifice and the relaxation regions. For the larger par-ticles (15–30 nm), most losses occur in the inlet region of thepressure-reducing orifice. Alternate designs for inlet orifice werealso considered and their transmission efficiencies were stud-ied. It was found that the inlet orifice with a conical diverg-ing section led to a better transmission efficiency for particlessmaller than 10. For particles in a range of 10–30 nm, the orificewith a conical converging throat exhibited a higher transmissionefficiency.

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