A Design Tool for Aerodynamic Lens Systems

We report in this article the development of a tool to design and evaluate aerodynamic lens systems: the Aerodynamic Lens Calculator. This Calculator enables quick and convenient design of aerodynamic lens systems based on the parameterized knowledge gained from our detailed numerical simulations of flow and particle transport through aerodynamic lens systems. It designs the key dimensions of a lens system: pressure limiting orifice, relaxation chamber, focusing lenses, spacers and the accelerating nozzle. It also provides estimates of particle terminal axial velocities, particle beam width, and particle transmission efficiencies. This article describes in detail the information that is used in the design tool, including equations for pressure drop through orifices, contraction factors as a function of Reynolds, Mach and Stokes numbers, spacer length as a function of Reynolds number and estimations for the relaxation chamber dimensions. The article also evaluates the performance of the design tool by comparing its predictions with the performance of aerodynamic lens systems that have been described in the literature.

[Supplementary materials are available for this article. Go to the publisher's online edition of Aerosol Science and Technology for the following free supplemental resources: An instruction manual for the Aerodynamic Lens Calculator.]

NOMENCLATURE

A f	=	cross sectional area of an orifice
c =	=	speed of sound in the carrier gas
C c	=	Cunningham slip correction factor
C d	=	orifice flow discharge coefficient
D	=	particle diffusion coefficient
d B,i	=	beam diameter downstream of lens i
d f	=	orifice diameter
d n	=	nozzle diameter
d p	=	particle diameter
d p1	=	maximum particle diameter in the specified focusing size range
d pN	=	minimum particle diameter in the specified focusing size range
d r	=	minimum diameter of the relaxation chamber
d s	=	inner diameter of the spacer
d s,i	=	inner diameter of the spacer downstream of lens i
d t	=	step diameter of the accelerating nozzle
ER = 1/β = d s /d f	=	orifice expansion ratio
f(v pr )	=	distribution function of particle terminal radial velocity
Kn	=	flow Knudsen number
Kn*	=	critical Knudsen number for continuum flow
Kn p	=	particle Knudsen number
h = (d s − d f )/2	=	orifice step height
L	=	distance from the detector to the nozzle
l a	=	approach length
l e	=	pipe flow entrance length
l f	=	redevelopment length
l r	=	reattachment length
l s	=	length of spacers
l s,i	=	spacer length between lenses i and i + 1
L r	=	length of the relaxation chamber
L t	=	step length of the accelerating nozzle
[mdot]	=	mass flowrate
M	=	molecular weight of the carrier gas
Ma	=	Mach number
Ma*	=	critical Mach number for orifice flow to be choked
p 1	=	pressure upstream of an orifice
p 2	=	fully recovered pressure downstream of an orifice
p focusing	=	pressure upstream of a lens for focusing particles of a given size
p Ma	=	minimum pressure for flow to be subsonic
p max	=	maximum operating pressure of an aerodynamic lens
p Kn	=	minimum pressure for flow to be continuum
Q	=	volumetric flowrate
Q i	=	volumetric flowrate at stage i
R	=	universal gas constant
Re	=	flow Reynolds number based on orifice diameter
Re s	=	flow Reynolds number based on spacer diameter
r i	=	critical initial particle radial position
r pi	=	particle initial radial location in an aerodynamic lens
S	=	Sutherland constant
S a	=	particle axial stopping distance
S r	=	particle radial stopping distance
St	=	Stokes number based on orifice diameter
St o	=	optimum Stokes number
St s	=	Stokes number based on spacer diameter
Sts50	=	Spacer Stokes number corresponding to a 50% impaction loss
stage i	=	includes lens i and the downstream spacer
t	=	particle residence time
T 1	=	temperate upstream of the lens
T pF	=	particle frozen temperature in the jet expansion
T r	=	reference temperature in Sutherland's Law
u	=	average flow velocity at orifice entrance based on upstream flow conditions
u p	=	particle axial velocity
U s	=	average flow velocity in the spacer
v a	=	axial jet flow velocity
v pr	=	particle terminal radial velocity in the vacuum chamber
v r	=	radial jet flow velocity
x c	=	≈ 1 −
x rms	=	particle root mean square displacement due to diffusion
x rms, i	=	root mean square displacement between lenses i and i+1
Y	=	orifice flow expansion factor
β = d f /d s	=	constriction ratio
γ	=	specific heat ratio of the carrier gas
Δ p	=	pressure drop across an orifice
η c	=	particle contraction factor
η c,i	=	contraction factor at lens i
η t	=	particle transmission efficiency
η t, diffusion, i	=	penetration after diffusional loss at stage i
η t, GK, i	=	penetration after loss at stage i estimated by Gormley-Kennedy equation
η t, orifice, i	=	transmission efficiency after impaction losses on the orifice plate i
η t, spacer, i	=	transmission efficiency after impaction loss to spacer i
θ	=	jet opening angle
λ1	=	mean free path of the gas molecules upstream of the orifice
μ	=	carrier gas viscosity
ξ =	=	dimensionless diffusion deposition parameter
ρ1	=	carrier gas density upstream of the aerodynamic lens
ρ p	=	particle material density
τ	=	particle relaxation time

INTRODUCTION

Aerodynamic lens systems have been widely used to generate collimated narrow particle beams since they were invented by Liu et al. (1995a), (b). Typical applications include inlets for aerosol mass spectrometers (Ziemann et al. 1995; Schreiner et al. 1998, 1999, 2002; Jayne et al. 2000; Tobias et al. 2000; Öktem et al. 2004; Su et al. 2004; Svane et al. 2004; Drewnick et al. 2005; Zelenyuk and Imre 2005), nanostructured material synthesis (Girshick et al. 2000; Dong et al. 2004; Piseri et al. 2004), and micro-scale device fabrication (Di Fonzo et al. 2000; Gidwani 2003).

In the original design of the aerodynamic lens system (Liu et al. 1995a, (b), aerosols are sampled from one atmosphere pressure through a pressure limiting orifice into the lenses which operate at a pressure of ∼200 Pa, with low pressure drop across each lens. Typically a series of three to five lenses progressively focus spherical particles in the size range of 25–250 nm (near unit density) into a narrow beam. The collimated particle beam is then delivered through an accelerating nozzle to the detection chamber maintained at very low pressure. Most lens systems being used today are similar in design to the system described by Liu et al. (1995a), (b).

Several studies have reported on lens systems that were designed to operate under conditions that are significantly different from those used by Liu et al. (1995a), (b). For example, Di Fonzo et al. (2000) designed a lens system to focus silicon carbide particles in the diameter range of 10–100 nm using an argon-hydrogen mixture; Dong et al. (2004) used a lens system to focus 10–200 nm silicon particles in hydrogen. Schreiner et al. developed aerodynamic lens systems operating at elevated pressures (2.5–20 kPa) for stratospheric aerosol studies (Schreiner et al. 1998, 1999, 2002). Lee et al. (2003) used a single lens to focus micron-sized particles in the atmospheric pressure range. Wang et al. (2005a), (b) developed a systematic procedure to design aerodynamic lenses for 3–30 nm nanoparticles.

To characterize the performance of an aerodynamic lens system, detailed numerical or experimental studies are required. Liu et al. (1995a), (b) carried out theoretical, numerical, and experimental analysis of the performance of aerodynamic lens systems. Zhang et al. (2002), (2004) repeated the simulations by Liu et al. with a compressible flow model. To study particle loss and beam broadening due to diffusion, Gidwani (2003) and Wang et al. (2005b) simulated the flow and particle transport in the lens system considering the particle Brownian motion.

Although numerical simulations or experimental evaluation of the performance of lens systems can provide more accurate results, they are very costly in both time and financial resources. Furthermore, these studies require predefined dimensions and operating conditions of the aerodynamic lens system, which need to be iteratively optimized. Due to the wide applications of aerodynamic lenses, a simple tool that can provide quick and reasonably accurate lens design and evaluation would be useful. We report the development of such an Aerodynamic Lens Calculator in this article. We studied flow and particle transport through a single lens at a wide range of Reynolds and Mach numbers. From these simulations, we obtained information about particle contraction factors, transport efficiencies, flow discharge coefficients through orifices and the lengths of spacer for flow to redevelop. We also simulated flow through the pressure limiting orifice and the relaxation region, the accelerating nozzle, as well as the entire lens system (Wang 2005b). Results from these detailed numerical calculations were parameterized and incorporated into the design tool described in this paper. We correlated the particle terminal axial velocity in the vacuum chamber downstream of the nozzle as a function of Stokes number with a simple equation. This enabled us to estimate the particle terminal axial velocity under similar nozzle geometry and operating pressure. We also provide estimate of the particle beam width both inside and downstream of the lens system. Both aerodynamic focusing and diffusion broadening are taken into account in the beam width estimation. We considered three particle loss mechanisms to calculate the particle transport efficiency: inertia impaction on the orifice plate, impaction on the spacer walls due to defocusing, and losses due to diffusion.

This design tool avoids the need for the time consuming detailed numerical simulations, and enables designers to quickly test a range of design options. The tool calculates the lens dimensions, operating conditions and flow parameters at each lens stage. It also estimates the terminal particle velocity after the choked accelerating nozzle, particle beam diameter at each stage and at the location of the detector, and particle transport efficiency through the lens system. This article describes the detailed information used in the Calculator. We first describe the analytical, empirical or numerical relations used to design the dimensions and operating conditions of aerodynamic lens systems. Next we describe the method used to estimate the particle terminal velocity, beam width and transport efficiency. Finally, we report the validation of this lens design tool by comparing its predictions with several detailed numerical and experimental studies reported in the literature.

AERODYNAMIC LENS SYSTEM DESIGN

The objective of aerodynamic lens design is to build a lens system that has best performance, i.e., maximum particle focusing, minimum particle losses, and minimum pumping requirements under user specified inputs (particle size range, particle density, etc.). To achieve this goal, one needs to optimize dimensions of the lens system (number of lenses, lens diameters, inner diameter and length of the spacers between two lenses, the flow limiting orifice, relaxation chamber and the accelerating nozzle) and operating parameters (flowrate, pressure, and carrier gas).

We have briefly described the aerodynamic lens design procedure in an earlier article with emphasis on lenses for nanoparticles (Wang et al. 2005a). In this section, we describe the design principle in more detail, and generalize the guidelines to be applicable to particles ranging in size from several nanometers to greater than one micrometer.

Assumptions

Figure 1 shows a schematic of an aerodynamic lens system with the nomenclature used in this paper and in the code of the lens calculator. The lens system described in this article consists of the following parts: an inlet orifice followed by a relaxation chamber, one or more lenses separated by spacers, and an accelerating nozzle. The pressure limiting orifice determines the volumetric flowrate through the lens system, and the accelerating nozzle defines the operating pressure. We assume that the lenses are primarily responsible for all focusing. The focusing or defocusing effects that might be caused by the inlet orifice or the accelerating nozzle are not controlled by the design tool, but the user can optimize these effects by varying the operating pressure. The inner diameters of each spacer are usually the same so as to simplify machining.

FIG. 1 Schematic and nomenclatures of the lens system.

Particles are assumed to be spherical and electrically neutral. The effects of particle shape have been discussed by Liu et al. (1995a), (b) and Huffman et al. (2005). The design tool restricts flow through the lenses to be laminar, subsonic, and continuum.

It is well known that flow instability and turbulence can disperse particles and destroy focusing. However, the Reynolds number above which these effects contribute to defocusing is not known with certainty. Eichler et al. (1998) and Gómez-Moreno et al. (2002) reported that turbulent transition in an orifice flow occurred at Reynolds number around 70. Back and Roschke (1972) and Gong et al. (1996) found flow instabilities around a Reynolds number of 200 downstream of sudden expansions. Our design tool constrains Reynolds numbers (Re) to be below 200. Therefore,

Although sonic or supersonic nozzles have been widely used to focus particles (Murphy and Sears 1964; Israel and Friedlander 1967; Cheng and Dahneke 1979; Dahneke and Cheng 1979; Fernández de la Mora et al. 1989; Mallina et al. 2000; Tafreshi et al. 2002), we constrain the flow through lenses to be subsonic to avoid the complicated influence of shock waves on particle focusing. Therefore, the lens Mach number (Ma) is limited to be smaller than the value corresponding to choked flow (Ma*) (Wang et al. 2005a), i.e.,

If the flow is in the free molecular regime, the particle inertia would greatly exceed the drag force and no focusing could be achieved. Furthermore, rarefied gas dynamics is too complicated to handle using our simple design tool. Therefore, we restrict the flow Knudsen number (Kn) to be smaller than 0.1 to assume that the flow is continuum,

Focusing Lens

The parameter that governs particle focusing through an aerodynamic lens is the Stokes number (St), which is defined as the ratio of the particle stopping distance at the average orifice velocity (u) to the orifice diameter (d _f ):

where C _c = 1 + Kn _p (1.257 + 0.4e and Kn _p = 2λ₁/d _p . As shown in earlier studies, the particle contraction factor (η _c ), which is the ratio of terminal to initial radial positions of the particles travelling through the lens, is a strong function of the Stokes number: |η _c | ≈ 1 when St ≪ 1, | η _c | < 1 when St ≈ 1, and | η _c | > 1 when St ≫ 1. In other words, small particles follow gas streamlines and thus are not focused, intermediate-sized particles are focused and large particles are defocused by the lens. There exists an optimum Stokes number (St _o ) for which η _c = 0 (Liu et al. 1995a). Liu et al. showed that the contraction factors are independent of the particle initial radial position for particles that are not too far away from the axis. We refer to these near-axis contraction factors unless stated otherwise.

By rearranging Equation (4), we can calculate the diameter of the lens aperture for optimally focused particles:

Note that the optimal Stokes number is a function of flow Reynolds number, Mach number, and lens geometry (Liu et al. 1995a; Zhang et al. 2002). We use thin plate orifices as the focusing elements in this paper. Liu et al. (1995a) showed that the dependence of St _o on the constriction ratio (β) of thin plate orifices becomes weak when β ≤ 0.25. To simplify the design process, we use β ≤ 0.25 in the design tool. In principle, one could use β > 0.25 if the η _c vs. St curves were known.

To obtain the relation between St _o , Ma, and Re, we have carried out numerical simulations of particle motion through a single lens at a range of Mach number (Ma = 0.03–0.32) and Reynolds numbers (Re = 1–100) which cover the typical operating conditions of lenses. In these simulations, β was set equal to 0.2. Particles were injected at an initial radial location (r _pi ) of 0.15d _s from the lens axis and Brownian motion was neglected. The relationship between the contraction factor and St, Re, and Ma is shown in Figure 2(a)–(c). Note that the dependence of contraction on Reynolds number decreases as Mach number increases. The dependence of the contraction factor on the particle initial radial location becomes significant for large Stokes numbers (Liu et al. 1995a). Therefore, the near-axis contraction factors in Figure 2 are not very representative at the higher St end of each curve, and significant impaction losses occur for particles with St > 10. Figure 3 shows the optimum Stokes number St _o as a function of Re for three different Ma, which corresponds to the St where η _c = 0 at each curve in Figure 2. The value of St _o for a specific set of Re and Ma can be interpolated from this data.

FIG. 2 Near-axis particle contraction factor as a function of the Stokes number for various Reynolds numbers at three different subsonic Mach numbers. (a) Ma = 0.03; (b) Ma = 0.10; (c) Ma = 0.32.

FIG. 3 Optimum Stokes number as a function of Reynolds number for three different Mach numbers.

From Equation (5), we can see that the pressure upstream of the lens is an important parameter in calculating the lens diameter. Therefore, we need to accurately estimate the pressure drop across an orifice. We have developed a model for viscous flow through an orifice

as well as for the discharge coefficient (Wang et al. 2005a)

The expansion factor Y is calculated using the expression of (Bean 1971):

Using Equations (5, 6, 7, 8) we can calculate the diameter of each lens for a given mass flowrate and a pressure at any stage of the lens assembly.

To study the particle size range that a multiple lens system can focus, we found from Equation (4) that in the free molecular regime St is proportional to d _p while in the continuum regime it is proportional to d ² _p . If a lens system focuses particles of a decade size range d _p1–d _pN with d _p1 = 10 d _pN , then the Stokes number of d _p1 at the last lens will be 10St _o if it is in the free molecular regime, and 100St _o if it is in the continuum regime. If by any non-ideal effects particles of d _p1were not focused exactly onto the axis, they will start to defocus at the last lens or even at an earlier stage. Therefore, it is probably a good idea to design a lens system to cover approximately a decade of particle size.

Substantial focusing can be achieved even with sub-optimal Stokes numbers when a sufficient number of lenses are used. There are two strategies to focus a wider range of particle sizes. First, one can add additional lenses to refocus the larger particles before they are defocused and lost. Second, one can use first several lenses to focus the largest particles (d _p1) to a given tolerance close to the axis, and then design the following lenses at η _c (d _p1) = − 1 and focus smaller sizes sub-optimally.

If particles are large enough that diffusion is not significant, and the largest particles do not defocus in the final stages, the more lenses used in an assembly, the better focusing can be achieved. In practice, however, increasing the number of lenses increases the difficulty of alignment and adds size and expense. Effects of diffusion increase as the lens length increases. Furthermore, larger particles are more prone to be defocused in an assembly with more lenses. Typically, we have found that a lens assembly consisting of five lenses is sufficient to focus particles of a decade size range. Three or four lenses were used in our previous study of nanoparticle focusing (Wang et al. 2005b).

Operating Pressures

From Equation (4), we obtain the pressure (p _focusing ) for focusing particles d _p using the ideal gas law

Note that C _c is a function of p _focusing . The minimum pressure for flow to be subsonic (p _Ma) and continuum (p _Kn) are respectively given as (Wang et al. 2005a)

and

Figure 4 shows examples of p _Ma, p _Kn, and the required p _focusing to focus particles of different sizes as functions of the orifice size in Figure (4a) air and Figure (4b) helium when the flowrate is 0.1 standard liter per minute (slm). The lens operating pressures are those along the p _focusing curves where p _focusing is wider than both p _Ma and p _Kn. Note that larger particles have a larger operating pressure window, and that using a lighter carrier gas can increase the operating pressures for smaller sizes which enables focusing them. Also shown in Figure 4(a) is a “Re warning line,” which corresponds to Re = 200 above which turbulence is likely to degrade focusing.

FIG. 4 The operating pressures for focusing unit density particles of different sizes as a function of orifice size using (a) air and (b) helium as the carrier gas. The flowrate is 0.1 slm. The two dash lines are the lower pressure limits p _Ma and p _Kn, respectively. The solid lines are p _focusing for indicated sizes.

As was pointed out previously (Wang et al. 2005a), increasing the operating pressure can reduce the detrimental effects of nanoparticle diffusion and can reduce the pumping needs. From Figure 4, we can see that the maximum operating pressure of smaller sizes occurs where the p _Ma curve intersects the p _focusing curve for each particle size, and its value can be derived from Equations (9) and (10),

Note that this is an implicit expression because C _c is a function of p _max, and an iterative method is needed to calculate p _max. The p _Ma curve may not intersect the p _focusing curve for larger particles in the practical orifice diameter range. In this case any value of p _focusing larger than p _Kn can be used. However, Figure 4 shows that the higher the operating pressure, the larger is the Reynolds number. It follows that Reynolds numbers tend to limit the maximum pressure at which lenses can be operated.

Length of Spacers

Spacers are used to separate lenses. They provide room for generating the periodic converging/diverging flow pattern with orifices, which drives aerodynamic focusing. In addition to the constraints mentioned earlier (β ≤ 0.25; equal inner diameter of all spacers), the spacers should be long enough so that the flow from the preceding lens can relax back to fully developed pipe flow before reaching the next lens (Liu et al. 1995b). Typically, spacer lengths are 10–50 times the upstream orifice diameter depending on the orifice Reynolds number (Wang et al. 2005a). However, no quantitative guideline on the spacer length has been reported.

Figure 5 shows the flow field through two lenses separated by a spacer. The flow separates from the wall when it passes through the lens and a recirculation zone forms downstream of the orifice. After a distance l _r (reattachment length), the flow reattaches to the wall, and becomes fully developed at a distance of l _f (redevelopment length) downstream of the orifice. The existence of the downstream lens is felt by the flow l _a (approach length) upstream of the next lens where the flow starts to curve toward the centerline. Ideally, the minimum length of spacers (l _s ) should be l _s1 = l _r + l _a , and more safely, l _s2 = l _f + l _a . If the spacer length is less than l _s1, the recirculation zone will likely fill the whole length of the spacer and less focusing will occur. Furthermore, particles are more likely to be entrained in the recirculation zones and be lost to the wall. Therefore, when particle diffusion is negligible and the overall length of the lens system is not a concern, we suggest using spacer lengths at least l _s2. When diffusion is important, l _s1 < l _s ≤ l _s2 can be used.

FIG. 5 Streamlines through two lenses separated by a spacer.

To estimate the values of l _r , l _f , and l _a , we have carried out numerical simulations of flow through single lenses in the Reynolds number range of 0.1–200. Calculated values for l _r are shown in Figure 6, together with data provided by other researchers (Back and Roschke 1972; Oliveira et al. 1998). Note that our numerical results are very close to those by Oliveria (1998), which apply to similar Reynolds numbers and expansion ratios. The results by Back et al. differ somewhat from ours and those of Oliveria, probably due to the difference in ER. Since we did not carry out simulations for Re > 200, we will use the data by Back et al. to estimate l _r in this Re range, noting that applying Back's data (ER = 2.6) to larger ER values might lead to some error. Also shown in Figure 6 is a fitted curve to the whole Reynolds number range, which can be described as follows

FIG. 6 Reattachment length of flow downstream of an axisymmetric expansion.

The redevelopment length l _f can be estimated by l _f = l _r + l _e where l _e is given as (Young et al. 2000)

The approach length l _a is found in our simulation to be independent of Reynolds number (Re < 150) with

Since l _a is relatively shorter than l _r and l _f , we use the above equation to estimate l _a for the whole Reynolds number range.

Flow Limiting Orifice, Relaxation Chamber, and Accelerating Nozzle

The flow limiting orifice, relaxation chamber, and the accelerating nozzle are important components of the aerodynamic lens system. The flow limiting orifice and relaxation chamber are required if the particle source pressure is higher than the lens operating pressure. The orifice sets the volumetric flowrate through the lens system, and reduces the pressure to the lens operating pressure. Usually the flow through the flow limiting orifice is critical. Therefore, a relaxation chamber is needed to slow down the high velocity flow and particles to prevent particle impaction on the downstream lens. The relaxation chamber also provides space for the flow to reattach to the wall after the recirculation eddies downstream of the orifice. The accelerating nozzle controls the exact lens operating pressure, and it accelerates particles to a downstream destination with minimized divergence angles.

The inner diameter of the relaxation chamber can be estimated from the radial stopping distance of the largest particles of interest. The lower half of Figure 7(a) shows the flow streamlines downstream of a straight-bored critical orifice with a diameter and wall thickness of 0.1 mm. The inlet pressure is 1 atm and the outlet pressure is 266 Pa. The flowrate is 70.6 sccm, which correspond to a Reynolds number of 1071 based on the orifice diameter. Therefore, the jet is turbulent. However, for the sake of simplicity, we only used a steady laminar flow model in this simulation and hence the streamlines in Figure 7(a) only serve as a qualitative description of the averaged flow behavior. The upper half of Figure 7(a) shows the half jet opening angle θ, which is defined as θ = tan ^{− 1}(h/l _r ) ≈ tan ^{− 1}(v _r /v _a ). Assuming v _a equals the speed of sound c, then v _r ≈ c tan (θ). We further conservatively assume that particle velocity equals to the flow velocity, then the particle radial stopping distance can be estimated from S _r = τ v _r where τ is the relaxation time of the largest particles in the target focusing size range downstream of orifice. The minimum diameter of the relaxation chamber (d _r ) is then

FIG. 7 Straight bored critical orifice and cylindrical relaxation chamber. (a) Flow streamlines and illustration of the half jet opening angle; (b) Trajectories of 100 nm (above axis) and 1 μ m (below axis) particles.

Here d _f is the diameter of the critical orifice.

The relaxation chamber should be long enough to let flow reattach and particles slow down. The length l _f for flow to redevelop can be calculated using Equations (13) and (14). Due to the complicated shock structure, we assume that particles have velocities equal to the speed of sound c for the carrier gas at a distance l _r downstream of the orifice while the gas velocity is negligibly low. The particle axial stopping distance downstream of l _r is then S _a = τ c. Therefore, the length of the relaxation chamber L _r can be estimated as

The shape of the relaxation chamber and critical orifice aperture are also important to reduce particle losses. Figure (7b) shows trajectories of 100 nm (above the axis) and 1 μ m (below the axis) particles through the critical orifice. The turbulent dispersion of particles is not included in the simulation. Note that some particles of both sizes are trapped inside the recirculation region for a long time, which makes them susceptible to coagulation or deposition losses. Therefore, we recommend that the strong recirculation region be eliminated. This can be achieved by inserting a conical expansion downstream of the orifice as shown in Figure 8(a). Figure 8(b) shows trajectories of 100 nm and 1 μ m particles through this modified relaxation chamber. Although this new design eliminates the recirculation eddy, impaction losses for larger particles increase due to the narrowed flow path, as is illustrated by the 1 μ m trajectories. To reduce impaction losses of larger particles, one can use conical nozzles, short capillaries or their combinations instead of thin plate orifices, since these nozzle shapes will reduce particle accelerations at the nozzle inlet (Fernández de la Mora and Riesco-Chueca 1988; Fernández de la Mora et al. 1989). Figure 8(c) illustrates that impaction losses of 1 μ m particles are reduced by adding a 0.1 mm thick 45° chamfer to the original orifice. (Note that this orifice modification increases the flowrate by 10.4%.) Although we demonstrated methods to improve the performance of the relaxation region in this section, we should point out that more careful studies of the orifice shape and chamber dimensions are still required.

FIG. 8 Modified relaxation chamber with a conical divergent section to reduce recirculation and particle loss. (a) Flow streamlines (straight orifice); (b) Trajectories of 100 nm (above axis) and 1 μ m (below axis) particles (straight orifice); (c) Trajectories of 100 nm (above axis) and 1 μ m (below axis) particles (orifice with a chamfer).

We follow the recommendation of Liu et al. (1995b) to use a step nozzle as the accelerating nozzle of lens systems (see Figure 1). The diameter of the final orifice in the nozzle can be calculated using Equation (6), and other dimensions of the nozzle are taken as d _t ≈ 2d _n , L _t ≈ d _s . Again, further research is needed to optimize the nozzle design so that the nozzle can also achieve maximum focusing for a given size range and operating conditions.

LENS PERFORMANCE ESTIMATION

The most important performance parameters for an aerodynamic lens system are: particle terminal velocities, particle beam widths at various locations, and the particle transmission efficiencies. These parameters need to be evaluated by detailed numerical simulations, and ultimately by experiments. However, it is attractive to have a best estimation of the lens performance with a “one-button-click” effort during the design process. In this section, we describe the method used in the Lens Calculator to do the estimations.

Estimation of Particle Terminal Velocities

We assume that the pressure downstream of the accelerating nozzle is low enough so that the gas-particle collisions are negligible and particles achieve their “terminal” velocities. We further assume that particles are in thermal equilibrium with the carrier gas upstream of the nozzle exit and their radial velocity can be approximately described by the Maxwell-Boltzmann distribution. This velocity distribution can be assumed to be frozen during the expansion downstream of the nozzle (Liu et al. 1995a; Wang et al. 2005b). Therefore, the terminal radial velocity v _pr can be estimated as

The particle terminal axial velocities (u _p ) depend on particle size, shape, nozzle geometry, pressure ratio, and carrier gases (Cheng and Dahneke 1979; Dahneke and Cheng 1979; Mallina et al. 1997). Figure 9 summarizes the particle terminal axial velocities for the four aerodynamic lens systems listed in Table 1. These four lens systems differ in dimensions, flowrate, pressure, and carrier gas. But they all use step accelerating nozzles similar to that described by Liu et al. (1995b), and the pressures downstream of the nozzles are less than 1 Pa. Note that the following equation fits the data points reasonably well in the St = τ c/d _n range where data is available (0.01 < St < 80)

FIG. 9 Normalized particle terminal axial velocities downstream of several aerodynamic lens systems.

TABLE 1 Key features of four lens assemblies with step accelerating nozzles used in particle axial velocity comparison

Lens system	d n (mm)	d s (mm)	d t (mm)	L t (mm)	Gas	p 1 (Pa)	d p (nm)
A (Wang et al. 2005b)	2.76	10	6	5	He	528	3–30
B (Liu et al. 1995b)	3.0	10	6	10	Air	93–333	25–250
C (TSI AFL-050)	3.2	17.78	7.62	17.78	Air	250	50–500
D (TSI AFL-100)	1.48	17.78	7.62	17.78	Air	600	100–3000

We will use this equation to estimate particle axial velocity in the Lens Calculator.

Estimation of Particle Beam Widths

The particle beam width is defined as the beam diameter that encloses 90% of the total particle flux. In this section expressions are given that are used in the Calculator to calculate particle beam widths downstream of each lens as well as downstream of the accelerating nozzle. Bean widths are controlled by two factors: aerodynamic focusing and diffusion broadening. The focusing can be inferred from Figure 2, and the diffusion broadening can be estimated by the root mean square displacement x _rms = , where D is the particle diffusion coefficient, and t is the particle residence time.

Assuming the diameters of spacers upstream and downstream of the pressure limiting orifice (lens 0) are the same, the flow is fully developed upstream of the pressure limiting orifice, and particles are homogenously distributed in the flow cross section, the starting particle beam diameter enclosing 90% flux is then ∼0.827d _s,0. The particle beam diameter before reaching lens 1 is

The beam diameter d _B,i at stage i inside the lens system can be easily calculated as

However, we should note that d _{B, i} may exceed the spacer diameter d _s,i due to defocusing or diffusion. In that case, we assume all particles outside the beam diameter of d _s,i are lost and reset d _B,i = 0.827 d _s,i .

Assuming particles achieve a Maxwell-Boltzmann distribution of radial velocities downstream of the acceleration nozzle (lens n + 1), we can estimate the particle beam width at a distance L downstream of the nozzle as follows:

Note that η _{c,n + 1}is only a very rough estimate because the pressure downstream of the nozzle is so low that the definition of contraction factor is no longer valid.

Estimation of Particle Transmission Efficiencies

Particle losses in an aerodynamic lens system arise from impaction and diffusion. Particle impaction happens both on the orifice plate and on the spacer walls.

We can view the orifices (including the pressure limiting orifice, lenses and the nozzle) as an impactor plate. Particles with large Stokes numbers will fail to follow streamlines and will impact on the plate. Therefore, it is appropriate to use the spacer Stokes number (St _s ) to characterize particle impaction on the orifice plate. The characteristic velocity and length in St _s are the average flow velocity in the spacer (U _s ) and the spacer inner diameter (d _s ), respectively, i.e., St _s = τ U _s /d _s , where U _s = 4 Q/(π d ² _s ). Figures 10(a), (10b), (10c) show the particle transmission efficiency as a function of St _s , Re and Ma for the same conditions as those in Figure 2. The corresponding Stokes numbers based on orifice are also shown in the figures. Note that significant particle losses happen in the St _s range of 0.1–1 for all Reynolds and Mach numbers. Although losses in these figures consist of both losses on the orifice plate and the downstream spacer walls, examination of particle trajectories shows that most losses are due to impaction on the orifice plate for these particular simulations. The transmission curves asymptotically reach η _t = for very large Stokes numbers corresponding to geometrical blocking of the orifice plate (Zhang et al. 2002). Figure 11 shows the cutoff Stokes number (St_s50) at which the transmission efficiency is 50% as a function of Re for the three Ma's studied. From Figure 10 we can see that the impaction particle loss is relatively steep. Therefore, the particle transmission efficiency through the stage from lens i–1 to lens i only considering impaction losses on the orifice plate i, η _{t, orifice, i} , can be estimated as a step function:

FIG. 10 Particle transmission efficiency as a function of the Stokes number for various Reynolds numbers at three different subsonic Mach numbers. (a) Ma = 0.03; (b) Ma = 0.10; (c) Ma = 0.32.

FIG. 11 Cutoff Stokes numbers as a function of Reynolds number for three different Mach numbers.

Note that non-uniform particle concentration due to focusing by upstream lenses is taken into consideration in this equation.

Defocused particles will be lost to the wall when η _c < − 1. In this case, the initial particle radial position upstream of the lens corresponding to the limiting trajectory of particle loss to the downstream spacer is r _i = d _s,i /(2η _c,i ). The transmission efficiency can be estimated as the ratio of particle flux within r _i to the total flux in that stage. Therefore,

Note that d _B,i = d _{B,i − 1}× |η _c,i |, and the diffusion effect is neglected.

Loss by diffusion at each stage can be estimated through the Gormley-Kennedy equation (Gormley and Kennedy 1949):

Note that we have assumed fully developed laminar flow in the lens system by using this equation. The actual diffusion loss is less than that predicted by the Gormley-Kennedy equation, because lenses move particles away from walls (Wang et al. 2005b). Therefore, the actual penetration accounting for diffusion falls somewhere between the Gormley-Kennedy prediction and 1 when particles are focused to a certain degree. We use the following simplified equation to predict diffusional loss:

The total particle transmission efficiency through the lens system is then

DESCRIPTION OF THE “LENS CALCULATOR”

A Microsoft Excel spreadsheet serves as a user interface for the Lens Calculator. The detailed calculations are done in the background by a program written in Visual Basic. The calculator has two modules: lens design and lens test. The lens design module designs lens dimensions and operating conditions for user specified parameters. The lens test module estimates the performance for a lens system with specified dimensions and operating conditions.

Figure 12 shows the flow chart of the lens design module. The design process starts with reading in the user input information of carrier gas, several operating parameters, particle density, and the focusing size range. Then the program calculates the operating pressure and dimensions of the lens system (orifice diameters and the length and inner diameter of spacers). Finally, the program estimates the three major performance parameters: size dependent particle terminal axial velocity, beam width, and transmission efficiency.

FIG. 12 Flow chart for the aerodynamic lens design module.

The user can either specify an operating pressure, or let the program design the lens system so that it operates at maximum possible pressure. The latter feature is especially useful when one tries to minimize the pumping capacity or the particle diffusion effects. The user can also let the program design lenses to operate at optimal or user specified Stokes numbers. The latter feature is typically used when focusing nanoparticles because sometimes it is not possible to operate lenses at optimal Stokes numbers while flow is subsonic and continuum (Wang et al. 2005a, b). It is also useful when designing lenses to operate at alternating optimal/suboptimal Stokes numbers to focus a wider size range. The flow through lenses is always forced to be laminar, continuum, and subsonic. Flow through the pressure limiting orifice can be turbulent and supersonic, but flow through the accelerating nozzle is forced to be choked because otherwise particles may have a short stopping distance and be lost to the pumps.

The lens test module is very similar to the design module except that the lens dimensions are specified by the user as well. The program calculates the operating conditions and estimates the assembly performance.

VALIDATION OF THE LENS CALCULATOR

In this section, we compare the performances (transmission efficiency and particle beam width) of several aerodynamic lens systems reported in the literature with predictions by the lens calculator. Three lens systems are selected in this comparison. Lens system 1 was designed by Wang et al. to focus 3–30 nm particles using helium as the carrier gas (Wang et al. 2005b); Lens system 2 was designed by Liu et al. and reported to have good focusing for 25–250 nm particles (lens e in Liu et al. 1995b); Lens system 3 was designed by Schreiner et al. to focus 0.34–4 μ m particles in the pressure range of 2.5–5 kPa (Schreiner et al. 1999, 2002). These three lens systems cover a wide range of focusing sizes and operating pressures. Only numerically simulated performance is reported for lens system 1, while experimental data are available for lens systems 2 and 3. The major features of these lens systems are listed in Table 2.

TABLE 2 Key features of the three aerodynamic lens systems used for the lens design tool validation

Lens system	d f (mm)	p 1 (Pa)	d p (nm)	Q STP (slm)
1 (Wang et al. 2005)	1.26, 1.64, 2.33, 2.76	528	3–30	0.1
2 (Liu et al. 1995)	5.0, 4.5, 4.0, 3.75, 3.5, 3	291	25–250	0.108
3 (Schreiner et al. 2002)	1.4, 1.2, 1.0, 0.8, 0.7, 0.6, 0.5, 0.4	5000	300–5000	0.052

Figure 13(a) compares reported particle penetrations through the above mentioned lenses with results predicted by the lens calculator. Note that the Calculator predicts particle penetration reasonably well for the three lens systems. For lens system 1, the prediction is lower than the numerical simulation in the size range of 1.5–40 nm. The first reason is, as we mentioned earlier, the diffusion loss estimation model in the Calculator is not very accurate (Equation 26). The second reason is that the particle penetration through the relaxation chamber was not included in the numerical simulation (Wang et al. 2005b) but it is included in the Calculator estimation. The Calculator also over predicts the particle size at which significant impaction losses occurs. This is most probably due to inaccuracies of interpolated contraction factors from Figure 2 and cutoff Stokes numbers from Figure 11. The discrepancies between the Calculator prediction and experimental data of the lens system are probably due to the nonideal factors in experiments. The Calculator prediction has a larger discrepancy with the experimental data for lens system 3. Two possibilities contribute to this difference. First, this lens system operates at larger Reynolds numbers (50–170), where flow instability might play a role. Second, the spacers between lenses are very short (15 mm) as compared to suggested values (∼ 20–50 mm). Therefore, recirculation will probably fill the whole spacer and cause extra losses.

FIG. 13 Comparison of the lens calculator prediction with experimental or numerical evaluation for three lens systems in the literature. (a) Transmission efficiency; (b) Particle beam width. The legends of the two figures are the same.

Figure 13(b) compares reported particle beam widths with values found using the Calculator. We can see that for lens system 1, agreement is very good for particles smaller than 10 nm. However, the deviation is significant for particles larger than 10 nm. The main reason is that the Calculator cannot predict the focusing/defocusing behavior of the accelerating nozzle very well. The agreement between the prediction and experiment for lens system 2 is quite good. Due to the fact that the design of lens system 3 did not closely follow the guidelines in this paper (shorter spacer, smaller orifice/spacer diameter ratio), the agreement between the prediction and measurement of beam width is only fair.

From the above comparisons, we can see that although there are some discrepancies between the prediction and reported data, the lens performance predicted by the Calculator is reasonable over a wide range of particle sizes and lens operating conditions.

SUMMARY AND CONCLUSIONS

A software package that is able to design and evaluate aerodynamic lens systems is reported in this article. This lens calculator has a design module and a test module. The design module calculates lens dimensions based on the user's inputs, and the test module evaluates the focusing performance for a lens system with specified dimensions and operating conditions. Empirical relations derived from numerical simulations or experiments were used in the lens design and performance estimation. Therefore the Calculator provides quick results with reasonable accuracy.

This article provided details of relationships used by the Calculator. These include the design of major components of the lens system: focusing lenses, spacers, the flow limiting orifice, the relaxation chamber, and the accelerating nozzle. We also showed graphically how to design the operating pressure of a lens system. Since a generalized expression of Stokes number that is applicable to both continuum and free molecular regimes is used and the effects of diffusion are taken into account, the lens Calculator applies to lenses for nanoparticles or micron sized particles.

Three performance parameters of a lens system are provided by the Calculator: particle terminal axial velocity, particle beam width, and particle transmission efficiency. A formula is used to correlate the terminal axial velocity to the particle Stokes number and speed of sound of the carrier gas. The particle beam diameter is estimated from contraction factor and the mean square displacement due to diffusion at each stage. Particle losses due to inertial impaction and diffusion are accounted for when calculating the transmission efficiency.

We compared the performance estimations by the lens calculator with the numerical or experimental results of three lens systems in the literature. The agreement is reasonably good.

Supplemental excel

Acknowledgments

This work was supported by NSF (Grant No. DMI-0103169) and the University of Minnesota Supercomputing Institute. We thank Dr. Frank Einar Kruis for helpful discussions.