Aerosol Science and Technology, 40:320–334, 2006
c American Association for Aerosol Research
Copyright ISSN: 0278-6826 print / 1521-7388 online
DOI: 10.1080/02786820600615063
A Design Tool for Aerodynamic Lens Systems
Xiaoliang Wang1,2 and Peter H. McMurry1
1
2
Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota, USA
Current address: TSI Inc., St. Paul, Minnesota, USA
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.]
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 presReceived 23 November 2005; accepted 3 February 2006.
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.
Address correspondence to Peter H. McMurry, 111 Church Street
SE, Minneapolis, MN 55455, USA. E-mail: mcmurry@me.umn.edu
320
sure 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. (1995 a, 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 argonhydrogen 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.
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS
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
FIG. 1.
321
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.
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
Schematic and nomenclatures of the lens system.
322
X. WANG AND P. H. MCMURRY
a Reynolds number of 200 downstream of sudden expansions.
Our design tool constrains Reynolds numbers (Re) to be below
200. Therefore,
Re =
ρ1 ud f
4ṁ
=
≤ 200.
µ
πµd f
[1]
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.,
Ma =
u
< Ma ∗ .
c
[2]
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,
Kn =
2λ1
< 0.1.
df
[3]
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 ):
St =
2ρ p d 2p Cc ṁ
τu
=
,
df
9πρ1 µd 3f
−1.1
[4]
where Cc = 1 + K n p (1.257 + 0.4e K n p ) and K n p = 2λ1 /d p .
As shown in earlier studies, the particle contraction factor (ηc ),
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: |ηc | ≈ 1 when St 1, |ηc | < 1 when St ≈ 1,
and |ηc | > 1 when St 1. In other words, small particles
follow gas streamlines and thus are not focused, intermediatesized particles are focused and large particles are defocused by
the lens. There exists an optimum Stokes number (Sto ) for which
ηc = 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:
df =
2ρ p d 2p Cc ṁ
9πρ1 µSto
13
.
[5]
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 Sto 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 ηc vs. St curves were known.
To obtain the relation between Sto , 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
0.15ds 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
Sto as a function of Re for three different Ma, which corresponds
to the St where ηc = 0 at each curve in Figure 2. The value of
Sto for a specific set of Re and Ma can be interpolated from this
data.
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
ṁ = A f = Af Cd Y
1−
β4
Cd Y
1−
β4
2ρ1 ( p1 − p2 )
p1
2M p
RT1 p1
[6]
as well as for the discharge coefficient (Wang et al. 2005a)

(Re < 12)
0.1373Re0.5




1.118
−
0.8873
×
Ln(Re) + 0.3953 × [Ln(Re)]2



− 0.07081 × [Ln(Re)]3
Cd =
+ 0.005551 × [Ln(Re)]4 − 0.0001581 × [Ln(Re)]5




(12 ≤ Re < 5000)



0.59
(Re ≥ 5000)
[7]
323
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS
FIG. 3. Optimum Stokes number as a function of Reynolds number for three
different Mach numbers.
The expansion factor Y is calculated using the expression of
(Bean 1971):
Y = 1 − (0.410 + 0.350β 4 )
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.
p
.
γ p1
[8]
Using Equations (5–8) 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 2p . If a lens system focuses particles of a
decade size range d p1 –d pN with d p1 = 10d pN , then the Stokes
number of d p1 at the last lens will be 10Sto if it is in the free
molecular regime, and 100Sto if it is in the continuum regime. If
by any non-ideal effects particles of d p1 were 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 ηc (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.
324
X. WANG AND P. H. MCMURRY
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 ( pfocusing ) for focusing particles d p using the ideal gas law
pfocusing =
2ρ p d 2p Cc ṁ RT
9πµd 3f M Sto
.
[9]
Note that Cc is a function of pfocusing . The minimum pressure for
flow to be subsonic ( pMa ) and continuum ( pKn ) are respectively
given as (Wang et al. 2005a)
pMa
ṁ c
=
Cd Yc A f
2M
xc
RT1
[10]
and
pKn =
T1 1 + S/Tr
2
λr pr
.
∗
df Kn
Tr 1 + S/T1
[11]
Figure 4 shows examples of pMa , pKn , and the required pfocusing
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 pfocusing curves where pfocusing is wider than both
pMa and pKn . 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.
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 pMa curve intersects the pfocusing
curve for each particle size, and its value can be derived from
Equations (9) and (10),
pmax
648ṁµ2 Sto2
=
πCd3 Yc3 ρ 2p d 4p Cc2
M
.
2RT xc3
[12]
Note that this is an implicit expression because Cc is a function
of pmax , and an iterative method is needed to calculate pmax .
The pMa curve may not intersect the pfocusing curve for larger
particles in the practical orifice diameter range. In this case any
value of pfocusing larger than pKn can be used. However, Figure 4
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 pMa
and pKn , respectively. The solid lines are pfocusing for indicated sizes.
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.
325
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS
FIG. 5.
Streamlines through two lenses separated by a spacer.
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 lr (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 la (approach
length) upstream of the next lens where the flow starts to curve
toward the centerline. Ideally, the minimum length of spacers
(ls ) should be ls1 = lr + la , and more safely, ls2 = l f + la .
If the spacer length is less than ls1 , 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 ls2 . When diffusion is important, ls1 < ls ≤ ls2 can be
used.
To estimate the values of lr , l f , and la , we have carried out numerical simulations of flow through single lenses in the Reynolds
number range of 0.1–200. Calculated values for lr 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 lr 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

−1.016 × 10−6 Re3 + 3.150 × 10−4 Re2 + 8.407




× 10−2 Re + 0.3689

for Re ≤ 200
lr
= 9.096 + 3860.168 × exp − log10 (Re)

h

0.431


 × sin 2π log10 (Re) − 4.226 for Re > 200.
1.272
The redevelopment length l f can be estimated by l f = lr + le
where le is given as (Young et al. 2000)
le
=
dt
0.06Res
1
6
4.4Res
for 0 ≤ Res < 2300
for Res ≥ 2300
.
[14]
The approach length la is found in our simulation to be independent of Reynolds number (Re < 150) with
la / h ≈ 1.625.
[15]
Since la is relatively shorter than lr and l f , we use the above
equation to estimate la 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
.[13]
FIG. 6. Reattachment length of flow downstream of an axisymmetric
expansion.
326
X. WANG AND P. H. MCMURRY
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/lr ) ≈ tan−1 (vr /va ). Assuming va equals the speed of sound c, then vr ≈ c tan(θ). We
further conservatively assume that particle velocity equals to the
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 µm (below axis) particles.
flow velocity, then the particle radial stopping distance can be
estimated from Sr = τ vr 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 (dr )
is then
dr = 2Sr + d f .
[16]
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 lr downstream of the orifice while the gas velocity is
negligibly low. The particle axial stopping distance downstream
of lr is then Sa = τ c. Therefore, the length of the relaxation
chamber L r can be estimated as
L r = max(l f + la , lr + Sa ).
[17]
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 µm (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 µm 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 µm 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 (8c) illustrates that impaction losses
of 1 µm 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.
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 dt ≈ 2dn , L t ≈ ds . Again, further research is
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS
327
FIG. 9. Normalized particle terminal axial velocities downstream of several
aerodynamic lens systems.
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 vpr
can be estimated as
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 µm (below axis) particles (straight
orifice); (c) Trajectories of 100 nm (above axis) and 1 µm (below axis) particles
(orifice with a chamfer).
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.
2 m p vpr
mp
f (v pr )dvpr =
2π vpr dv pr .
exp −
2π kT p F
2kT p F
[18]
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/dn
range where data is available (0.01 < St < 80)
u p /c = (0.939 + 0.09St)/(1 + 0.543St).
[19]
We will use this equation to estimate particle axial velocity in
the Lens Calculator.
328
X. WANG AND P. H. MCMURRY
TABLE 1
Key features of four lens assemblies with step accelerating nozzles used in particle axial velocity comparison
Lens system
A (Wang et al. 2005b)
B (Liu et al. 1995b)
C (TSI AFL-050)
D (TSI AFL-100)
dn (mm)
ds (mm)
dt (mm)
L t (mm)
Gas
p1 (Pa)
d p (nm)
2.76
3.0
3.2
1.48
10
10
17.78
17.78
6
6
7.62
7.62
5
10
17.78
17.78
He
Air
Air
Air
528
93–333
250
600
3–30
25–250
50–500
100–3000
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 xr ms =
2D t, 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
∼0.827ds,0 . The particle beam diameter before reaching lens 1
is
d B,0 = 0.827ds,0 ηc,0 + xrms,0 .
[20]
The beam diameter d B,i at stage i inside the lens system can be
easily calculated as
d B,i = d B,i−1 ηc,i + xrms, i ,
i = 1 to n.
[21]
However, we should note that d B,i may exceed the spacer diameter ds,i due to defocusing or diffusion. In that case, we assume
all particles outside the beam diameter of ds,i are lost and reset
d B,i = 0.827 ds,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:
2kT p F L
= d B,n ηc,n+1 + 3.04
.
mp up
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 (Sts ) to characterize particle impaction
on the orifice plate. The characteristic velocity and length in
Sts are the average flow velocity in the spacer (Us ) and the
spacer inner diameter (ds ), respectively, i.e., Sts = τ Us /ds ,
where Us = 4 Q/(π ds2 ). Figures 10(a)–(c) show the particle
transmission efficiency as a function of Sts , 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 Sts 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
d
d
reach ηt = 2( dsf )2 − ( dsf )4 for very large Stokes numbers corresponding to geometrical blocking of the orifice plate (Zhang
et al. 2002). Figure 11 shows the cutoff Stokes number (Sts50 )
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,
ηt, orifice, i , can be estimated as a step function:
ηt, orifice, i =

1
for

 d f,i 4
d f,i ) 2
2 d
− d
s,i−1

2 ds,i−1 4 for
 d
2
B,i−1
ds,i−1
−
B,i−1
ds,i−1
Sts < Sts50
or
d B,i−1 ≤ d f,i
Sts ≥ Sts50
and
d B,i−1 > d f,i
.
[22]
[23]
Note that ηc,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.
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 ηc < −1.
In this case, the initial particle radial position upstream of the
d B,n+1
329
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS
FIG. 11. Cutoff Stokes numbers as a function of Reynolds number for three
different Mach numbers.
lens corresponding to the limiting trajectory of particle loss to
the downstream spacer is ri = ds,i /(2ηc,i ). The transmission
efficiency can be estimated as the ratio of particle flux within ri
to the total flux in that stage. Therefore,
ηt, spacer, i =

1

 

2
ds,i
ds,i−1 ηc,i
d
2
B,i−1
ds,i−1
2 −
2 −
4
ds,i
ds,i−1 ηc,i
d B,i−1 4
ds,i−1
for d B,i ≤ ds,i
for d B,i > ds,i
.
[24]
Note that d B,i = d B,i−1 × |ηc,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):
ηt, GK, i =

2/3
1 − 5.50 ξ + 3.77ξ
for ξ < 0.009
0.819 exp(−11.5 ξ ) + 0.0975 exp(−70.1 ξ ) . [25]

for ξ ≥ 0.009
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:
ηt, diffusion, i =
(ηt,G K ,i + 1)/2
ηt,G K ,i
for d B,i < ds , i
.
for d B,i ≥ ds,i
[26]
The total particle transmission efficiency through the lens system
is then
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.
ηt = ηt, orifice, n + 1 ×
n
[ηt, orifice, i × ηt, spacer, i × ηt, diffusion, i ].
i=0
[27]
330
X. WANG AND P. H. MCMURRY
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
FIG. 12.
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
Flow chart for the aerodynamic lens design module.
331
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS
TABLE 2
Key features of the three aerodynamic lens systems used for the lens design tool validation
Lens system
1 (Wang et al. 2005)
2 (Liu et al. 1995)
3 (Schreiner et al. 2002)
d f (mm)
p1 (Pa)
d p (nm)
Q STP (slm)
1.26, 1.64, 2.33, 2.76
5.0, 4.5, 4.0, 3.75, 3.5, 3
1.4, 1.2, 1.0, 0.8, 0.7, 0.6, 0.5, 0.4
528
291
5000
3–30
25–250
300–5000
0.1
0.108
0.052
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.
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
µm 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.
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
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.
332
X. WANG AND P. H. MCMURRY
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.
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.
NOMENCLATURE
Af √
c = γ RT1 /M
Cc
Cd
D
d B,i
df
dn
dp
d p1
d pN
dr
ds
ds,i
dt
ER = 1/β = ds /d f
f (vpr )
Kn
K n∗
Knp
h = (ds − d f )/2
L
la
le
lf
lr
ls
ls,i
Lr
Lt
ṁ
M
Ma
Ma ∗
cross sectional area of an orifice
speed of sound in the carrier gas
Cunningham slip correction factor
orifice flow discharge coefficient
particle diffusion coefficient
beam diameter downstream of lens i
orifice diameter
nozzle diameter
particle diameter
maximum particle diameter in the specified focusing size range
minimum particle diameter in the specified focusing size range
minimum diameter of the relaxation
chamber
inner diameter of the spacer
inner diameter of the spacer downstream
of lens i
step diameter of the accelerating nozzle
orifice expansion ratio
distribution function of particle terminal
radial velocity
flow Knudsen number
critical Knudsen number for continuum
flow
particle Knudsen number
orifice step height
distance from the detector to the nozzle
approach length
pipe flow entrance length
redevelopment length
reattachment length
length of spacers
spacer length between lenses i and i + 1
length of the relaxation chamber
step length of the accelerating nozzle
mass flowrate
molecular weight of the carrier gas
Mach number
critical Mach number for orifice flow to
be choked
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS
p1
p2
pfocusing
pMa
pmax
pKn
Q
Qi
R
Re
Res
ri
r pi
S
Sa
Sr
St
Sto
Sts
Sts50
stage i
t
T1
Tp F
Tr
u
up
Us
va
vpr
vr
2 γ γ−1
xc ≈ 1 − γ +1
xrms
xrms,i
Y
β = d f /ds
γ
pressure upstream of an orifice
fully recovered pressure downstream of
an orifice
pressure upstream of a lens for focusing
particles of a given size
minimum pressure for flow to be subsonic
maximum operating pressure of an aerodynamic lens
minimum pressure for flow to be continuum
volumetric flowrate
volumetric flowrate at stage i
universal gas constant
flow Reynolds number based on orifice
diameter
flow Reynolds number based on spacer
diameter
critical initial particle radial position
particle initial radial location in an aerodynamic lens
Sutherland constant
particle axial stopping distance
particle radial stopping distance
Stokes number based on orifice diameter
optimum Stokes number
Stokes number based on spacer diameter
Spacer Stokes number corresponding to
a 50% impaction loss
includes lens i and the downstream
spacer
particle residence time
temperate upstream of the lens
particle frozen temperature in the jet expansion
reference temperature in Sutherland’s
Law
average flow velocity at orifice entrance
based on upstream flow conditions
particle axial velocity
average flow velocity in the spacer
axial jet flow velocity
particle terminal radial velocity in the
vacuum chamber
radial jet flow velocity
particle root mean square displacement
due to diffusion
root mean square displacement between
lenses i and i+1
orifice flow expansion factor
constriction ratio
specific heat ratio of the carrier gas
p
ηc
ηc,i
ηt
ηt, diffusion, i
ηt, GK, i
ηt, orifice, i
ηt, spacer, i
θ
λ1
µ
ξ=
Di ls,i
Qi
ρ1
ρp
τ
333
pressure drop across an orifice
particle contraction factor
contraction factor at lens i
particle transmission efficiency
penetration after diffusional loss at stage
i
penetration after loss at stage i estimated
by Gormley-Kennedy equation
transmission efficiency after impaction
losses on the orifice plate i
transmission efficiency after impaction
loss to spacer i
jet opening angle
mean free path of the gas molecules upstream of the orifice
carrier gas viscosity
dimensionless diffusion deposition parameter
carrier gas density upstream of the aerodynamic lens
particle material density
particle relaxation time
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