Resolution of the radial differential mobility analyzer for ultrafine particles

Pergamon

J. Aerosol

SCI Vol. 27, No. 8, pp. 1179-1200, 1996 Copyright 84; 1996 Elsewer Science Ltd Prmted m Great Bntain. All nghts reserved 0021.X502/96 115.00 + 0.00

SOO21-8502(96)00036-5

RESOLUTION OF THE RADIAL DIFFERENTIAL MOBILITY ANALYZER FOR ULTRAFINE PARTICLES Shou-Hua Division

of Chemistry

Zhang

and Richard

C. Flagan*

and Chemical Engineering, California Institute Pasadena, CA 91125, U.S.A.

(First received 28 December

1995; and

of Technology

210-41,

in final form 6 March 1996)

Abstract-The resolution of the radial differential mobility analyzer (radial DMA) for particles in 3%60nm diameter range is probed through tandem radial DMA measurements employing identical radial DMAs. The observed broadening of the range of transmitted particles with decreasing particle Peclet number was shown to be consistent with Stolzenburg’s (1988, Ph.D. Thesis, University of Minnesota) model of diffusion broadening of the transfer function, although the broadening was somewhat greater than predicted. A similar, but smaller, deviation is seen in Stolzenburg’s data obtained using a cylindrical DMA. The enhanced broadening is thought to result from flow disturbances within the DMA. Diffusional deposition of particles in the radial DMA for two while electrophoretic particle losses in the different sheath flow rates correlated well with Pe-“‘, transition from high voltage at the outlet of the DMA to grounded tubing are shown to be independent of the particle Peclet number. The latter effect is, however, small for the radial DMA. Consistent with observations previously made using cylindrical DMAs, the voltage corresponding to the peak in the number concentration is slightly higher for the second DMA than for the first one. Q 1996 Elsevier Science Ltd This apparent decrease in mobility correlates with Pe -‘. Copyright

INTRODUCTION Particle formation from the vapor phase generates ultrafine particles in the atmosphere as a result of photochemical reactions, in combustors, and in a wide variety of particle synthesis systems. Measurements of particles much smaller than 1 pm in diameter are central to the efforts to understand the origins of such particles. A number of methods are available for measurement of particles larger than 100 nm in diameter. Size distribution of ultrafine particles (D, < 100nm) are measured primarily by differential mobility analysis. The predominant mobility-based instrument, the cylindrical differential mobility analyzer (DMA), was introduced by Knutson and Whitby (1975a, b) and commercialized by TSI. This instrument has served the aerosol research community admirably for more than two decades, but it suffers substantial diffusion losses of particles below about 10 nm diameter. Needs for measurements of smaller particles, to resolve shorter transients or lower number concentrations than those for which the original instrument is well suited, and to satisfy weight and size constraints associated with installing the instrument on aircraft have led a number of investigators to develop new versions of the DMA (Reischl, 1991; Winklmayr et al., 1991; Mesbah, 1994; Zhang et al., 1995; Russell et al., 1996). The original Knutson and Whitby instrument has been extensively characterized (Knutson and Whitby, 1975a, b; Kousaka et al., 1985, 1986; Stolzenburg, 1988), but less is known about the newer instruments. This paper examines the resolution of one of the new designs, the radial DMA (RDMA) developed by Zhang et al. (1995). The performance of the differential mobility analyzers is commonly characterized in terms of a transfer function which is defined as the probability that a particle of a given mobility will be transmitted through the DMA (Knutson and Whitby, 1975a, b; Liu and Pui, 1974). For sufficiently large particles, the transfer function closely approximates the ideal expression that was originally derived by Knutson and Whitby (1975a). At smaller

*Author

to whom correspondence

should

be addressed.

Fax: 818 568-8743.

1179

Tel.: 818 395-4383.

1180

S.-H. Zhang and R. C. Flagan

particle sizes, however, the transfer function is significantly broader than what the simple model predicts. Using a cylindrical DMA of the Knutson and Whitby (1974a) design, Kousaka et al. (1985) demonstrated experimentally that this broadening is appreciable for particles smaller than about 40nm, and hypothesized that the broadening resulted from Brownian diffusion in the analyzer column. In a later paper, Kousaka et al. (1986) compared the experimentally measured transfer function with numerical predictions, confirming Brownian diffusion as the cause for the broadening. Brownian diffusion also leads to deposition of fine particles in the entrance and exit regions of the DMA, lowering the overall transmission efficiency (Kousaka et al., 1986; Reineking and Postendorfer, 1986). Kousaka et al. (1986) demonstrated that relatively minor changes in the design of the Knutson and Whitby DMA outlet could produce marked increases in the overall transmission efficiency for ultrafine particles. Further improvements in DMA performance have been achieved with more radical design changes (Reischl, 1991; Winklmayr et al., 1991; Zhang et al., 1995). Replacing the narrow annular aerosol inlet with a tangential-flow inlet significantly reduced entrance region losses, while reduced residence times decreased diffusional broadening in these instruments. Zhang et al. (1995) and Mesbah (1994) developed DMAs in which particles are classified in a radial flow between parallel disk electrodes. These instruments also feature tangential aerosol inlets and particularly simple outlet geometries that reduce exit region losses. Thus, mobility classifiers are now available that extend the usable operating range to particle diameters of only a few nanometers. Although the degradation is not as severe as with earlier instrument designs, Brownian diffusion still broadens the transfer functions of these instruments. Stolzenburg (1988) performed a detailed analysis of diffusional effects in cylindrical DMAs of the Knutson and Whitby design, leading to an approximate analytical solution to the coupled diffusion/migration problem in the cylindrical DMA. That work provides a basis for analyzing the influence of diffusion on the performance of the newer DMA designs. Indeed, Mesbah (1994) has applied Stolzenburg’s analysis to a radial differential mobility analyzer. More recently, a broadened triangular transfer function has been proposed for deconvoluting DMA measurements of ultrafine aerosol particles (Stratmann et al., 1995). Fissan et al. (1995,1996) applied this approach in an examination of the diffusional degradation of the performance of several different types of DMAs. They clearly demonstrated the importance of incorporating the diffusional effects in characterizing differential mobility analyzers. RosselllLompart and de la Mora (1993) modeled diffusion in a cylindrical DMA in an effort to obtain the optimal DMA design with respect to diffusional broadening of the transfer function. They concluded that the optimal cylindrical DMA would have a column length that is comparable to the gap between the concentric electrodes of the classifier section, a condition that is not met by any of the DMAs in common use at the time of their study. They demonstrated that the essential features of their predicted transfer function were reproduced experimentally in the classification of molecular ions. In this paper, we extend the model of Stolzenburg (1988) to the Zhang et al. (1995) radial DMA which we have previously characterized only in terms of the ideal, non-broadened triangular transfer function. We compare the predictions to the experimentally determined transfer functions, and show that deviations from the predicted performance are relatively minor and are correlated with the flow parameters of the instrument.

THEORY The differential mobility analyzer classifies particles migration velocities vE in an electric field. The migration strength and the electrical mobility, i.e. vE

=

Z,E,

by virtue of their electrophoretic velocity is the product of the field

Resolution

of the radial

DMA

1181

where the electrical mobility is related to the particle diffusivity, 9, by

2,=+,

(2)

B

and np is the number of elementary charges, e the elementary unit of charge, kB the Boltzmann constant, and T the temperature. Particle motion through the DMA is determined by the gas motion that advects particles in the streamwise direction and the electrophoretic migration in the direction of the imposed electric field, approximately normal to the direction of gas flow. The gas motion through the DMA can be described in terms of the fluid stream function, +, which, in cylindrical coordinates for an incompressible flow, is defined in terms of the fluid velocity, u, by (Bird et al., 1960) urs----,

1 dll/ r

U,E

dz

---,

1W

(3)

r dr

The flow in the DMA can be characterized in terms of the stream function values that bound the various entering and exiting flows, i.e. aerosol inlet flow rate

- 27$$2 - $1) = Qa,

- 2rc(ti4 - I/~) = Qsh, sheath inlet flow rate - 27@4 - $3) = Qs, - 2+3

classified sample outlet flow rate

- $I) = Q,,

(4)

excess flow rate at exhaust

where Ic/1> $2 > r+G3 > $4 since the flow is in the - r direction. At steady state, Qa + Qsh = Qs + Qe. Figure 1 illustrates these limiting stream lines in the radial DMA. The flow of particles through the DMA deviates from that of the gas due to electrophoretic migration in the imposed electric field which can be described in terms of the electric flux function 4 defined analogously to the stream function by (5) Non-diffusing particles follow trajectories function (Knutson and Whitby, 1974a),

corresponding

F(r,z) = * + Z,C$ =

to a constant particle stream

*” [ru,dz - ru, dr]. s

(6)

Representative particle streamlines and electric field lines are shown in Fig. 1. Small particles deviate from the trajectories defined by equation (6) due to Brownian diffusion, although, in the absence of boundary effects, this expression still describes the mean motion of the particles. The relative importance of particle migration and diffusion can be expressed in terms of the migration Peclet number for those particles that are transmitted through the DMA,

which is expressed in terms of the electrophoretic migration velocity since this form will appear naturally in the derivations that follow. Even expressed in terms of this relatively low velocity, migration Peclet numbers are generally large compared to unity, indicating that diffusional motions are small compared to migration. Nonetheless, diffusion does cause the particles to deviate from the deterministic trajectories. Stolzenburg (1988) modeled the diffusive migration of particles in the cylindrical, DMA in terms of a local, orthogonal curvilinear coordinate system based on streamwise and cross-stream coordinates, s and X,

S.-H. Zhang

1182

and R. C. Flagan

Streamlines,

Electric

Ground

m

Field

yf=constant

Lines,

/

%h

pconstant

?ectrode ?

Z

(C )

Fig.

1. Illustrations

the distance respectively.

Particle

of(a) flow streamlines,

Streamlines,

T=constant

(b) electrical field lines, and (c)particle radial DMA.

streamlines

in the

along the particle streamline Ti and the perpendicular distance from Ti, The relationship between these coordinates and the vectorial particle position r = re, + ze,

(8)

in the radial DMA is illustrated in Fig. 2. Several approximations were made in the derivation of the transfer function for diffusing particles. Because the particle Peclet numbers are large, only cross-stream diffusion (x-direction) was treated, i.e. diffusion in the streamwise direction was assumed to be negligible. Stolzenburg (1988) also assumed cross-stream shear to be negligible. Finally, particle losses to the walls of the classifier column were neglected. With these approximations, the root-mean-square displacement of particles from the mean trajectory increases with time according to the Einstein equation (Fuchs, 1964) d(.x2) = 2CZdt. (9) The distribution 0; = (x2).

of particles

about

the

mean

trajectory

is Gaussian

with

variance

Resolution

Fig. 2. Schematic

of the radial

DMA

illustrations of the broadening of the probability distribution particles in the rz plane of the radial DMA.

1183

of diffusional

The DMA transfer function is more conveniently derived in terms of the stream function than in terms of x. Since diffusion across a small portion of the flow channel can substantially distort the transfer function, it suffices to consider small deviations from the starting particle stream function, Ti. Defining x = 0 on the starting particle stream function, the variation of r with x can be approximated using a Taylor series (Stolzenburg, 1988).

r(X,S)Z ri where

Hence,

ar x=o + dx I Ix=oxy

(10)

ar dx z TV.

(11)

r(X)E ri + (n~)I,=~x.

(12)

The incremental change in the variance in particle stream function during a time interval dt can now be estimated, 2 do,2 E 2CBr2v2dt.

(13)

The total diffusional spreading experienced by particles during their transit through the DMA is, thus, exit a;

=

2gr2 v2 dt. 5e”tra*Ce

(14)

Particles that enter the classifier at stream function Ti are normally distributed about Ti at the classifier outlet. The probability that such a particle will exit between To and To + dTo is ftrans(ri,

row0

= &exp[---~(~~]dro.

(15)

The change in particle stream function lYcan be related to that for fluid stream function Ic/using equation (6) f-0 - Ti = $0 - $i +- &A$,

(16)

1184

S.-H. Zhang

and R. C. Flagan

where A@ is the change in the electric flux function DMA. For the radial DMA, this quantity is A$ = We may now express the probability

with migration

R; - R: 2h V.

of equation

To determine the overall diffusional all entering particles to the transmission and exiting flows, i.e.

associated

across the

(17)

(15) in terms of the fluid stream functions,

transfer function for the DMA, the contributions of must be determined by integration over all entering

(19) Equation (15) applies to both the radial DMA and the cylindrical DMA, so the diffusional transfer function for both geometries is that derived by Stolzenburg (1988), i.e.

(20) where the function

6?(x) is defined 8(x) Z?

The dimensionless

quantities

in terms of the error function,

Xerf(u)du s0 in equation

= xerf(x)

+ 1 exp[VL

erf(x), x2].

(21)

(20) are

(22)

(23)

(24)

(25) The centroid

mobility

for the ideal, non-diffusional z*

=

P

transfer

(Qsh+ Qe)b 2n(R;

- Rf)V

function

in the radial DMA is

(26)

To evaluate the broadened transfer function, we must first determine the dimensionless diffusional broadening parameter 5. Because or depends on X/ax, 6 is a function of both flow and electric fields in the classifier. The resolution of the DMA is limited by the ideal performance but may be reduced by diffusional broadening. To quantify these effects, we borrow terms from the mass spectrometry literature by defining the resolution as the ratio of the mobility corresponding to the

Resolution

of the radial

DMA

1185

peak in the transfer function to the full width of the transfer function at one-half of the maxtmum value, AZ,,,, 1.e. &+$

(27)

r/2

The ideal resolution limit is Bidear = p- I. For typical flow rates in the radial DMA, /I = 0.1 and 6 = 0.0, this leads to Bidear = 10. B has been evaluated numerically from equation (20) and is shown as a function of 5 in Fig. 3. To determine 5, we again follow Stolzenburg (1988), limiting our attention to cases where diffusional effects are small. As we shall see, even relatively minor diffusion can dramatically degrade the transfer function. To estimate c,- and 6, we examine, therefore, the influence of diffusion on particles that would be at the centroid of the ideal (non-diffusional) transfer functions, i.e. those that enter on streamline

(28) and have mobility 2:. For small diffusion, the mean stream function at which the particles will exit the DMA is (29) Hence, the range of stream functions crossed by the nondiffusing particles at the centroid of the transfer function is (30) The migration velocity for these particles is v;=Z*E

P

27

=

Qsh+ Qe

(31)

27c(R,2- R:).

10

8

6 R 4

Fig. 3. Resolution

of differential

mobility

analyzers as a function parameter 5.

of the dimensionless

broadening

1186

S.-H. Zhang

The characteristic

time for particle

and R. C. Flagan

migration

between

r=<.

the electrodes

is

h (32) UE

Defining

the dimensionless

quantities (33)

the dimensionless

variance

becomes 16Gb3

” =

(Qe + QsdWi - R:) s I-*

3r”2 d;=

G _ Pe’

(34)

where

+

pe=T= is the migration Peclet number transfer function. The influence sionless geometry factor,

b(Qsh Qe) 27r(R,2 - RI)2

(35)

of the aerosol particles at the centroid of the radial DMA of flow field and geometry is accounted for by the dimen-

,72 ,Y2 &

G”= 8F2

(36)

s I-*

where F = (I?; - Z?:)-‘. The detailed flow and electric fields must be known to determine G. The radial DMA has been designed to produce a uniform electric field, and the flow through most of the radial DMA is nearly parallel to the electrode surface. Hence, iYz2 6: = 1, leading to c” = 8g2 The integral

the integral

is evaluated

s

(1 + 6; )F’dtl I-*

along the trajectory

(37)

f = f*. Noting

that

becomes z; c” = 8&*

(1 + 6: )?‘(z”)dS,

(39)

s i: where the limiting values of z”correspond radial position with z”can be determined

to the streamlines from

di =- fi _=‘ dz” 6z

$T and $g. The variation

VI..

of the

(40)

The radial motion of the particle is determined by the mean fluid motion, i.e. 6, = C,. To determine the limits for integration, we examine the fraction of the total gas flow, F(T), between Z = 0 and .?, where z”= 0 is midway between the electrodes. This flow fraction is 1

‘(-2nuJrdz F(‘) = Qa + Qs,, s 0 The fraction

of the flow in the interval

=~~+.)r‘di.

(41)

0 < Z < ii* is (42)

Resolution of the radial DMA

while the fraction

in the interval

1187

0 < z”< z”$ is

(43) The specific values of z”T and z”$ for a particular velocity profile, u”,, are determined by equating equation (41) to (42) or (43), respectively. Particles migrate along particle streamlines corresponding to constant values of I-*. Expressed in nondimensional form, these trajectories in the radial DMA are described by

Since I’* is constant along the particle trajectory, we may equate results of equation (44) for radial positions r”and R2 to determine the radial position as a function of 2, leading to

l+P p = R”,2+ -F(q)--

l +pF(2)

E

Substituting

equations

R

(42) and (45) into equation

G = 8E2

R”,z- y

(45)

’

(39), we find,

1

- y

F(Z) + (Kr)2 dz.

(46)

We consider two limiting cases for the radial flow between the parallel disk electrodes: (i) fully developed laminar flow; and (ii) plug flow. Taking the origin of .? midway between the two electrodes, the radial velocity in the creeping flow limit (laminar flow) is (Bird et al., 1960) 31+/31-422 L&C = - -4E r”’

(47)

Assuming that the gas flow is strictly parallel to the electrodes, the quantity of z”only. In the creeping flow limit, the flow fraction becomes

4, ? is a function

F(Z) = ; z”- 2z3. Applying the trigonometric solved to yield

relationship,

where 0 G COS- ’ ( - 2F,) d 7~.Substituting

(48)

cos 39 = 4 cos3 0 - 3 cos 8, equation

(48) can be

equations

(46) we find

(47) and (48) into equation

1

-

&(l

+ /j)

z”;’ - z”T2 -

$2;’-

fT4)

1 + [8R2R”; - 4&l For 6 = 0, .?F = - Zg, and equation G, = 9(1 +/I)’

+ /3S)] (2; - z”T). (50) simplifies

(50)

to

1+

[16g2,;

- S@$

(51)

1188

S.-H. Zhang

For plug flow, the r&dial velocity

and R. C. Flagan

is 1+ p . l&p= - 2&r’

(52)

F,(F) = z.

(53)

and

By the method

outlined

Gp = [s&7: When

above.

the geometry

parameter

becomes

- 4E( 1 + PCs) + 2( 1 + /I)‘] (2; - 57) - 4R( 1 + b)(@

is = 0. 27 = - 5: which is given by equations 8E2R;

fq----

4K

1+ p

- 222).

(43) and (53) leading

to

+ 2(1 + p,.

1+p

(54)

(55)

The dimensions of the radial DMA of Zhang et ul. (1995) are: RI = 0.24cm, R2 = 5.04cm, b = l.OOcm. The instrument is normally operated with a flow rate ratio b = 0.1. These parameters lead to geometry parameters G, = 2.92 for creeping flow and G, = 2.34 for plug flow. The value of the dimensionless broadening parameter, 6 = J%%%, thus depends on the flow rate and on the particle diffusivity through the migration Peclet number. Table 1 summarizes these calculated diffusional broadening parameters for a variety of particle sizes for two different flow rates in the radial DMA for different value of ci. Figure 4 Table

1. Predicted

broadening

parameters

for the RDMA

(Qsh = Q., Q. = Q?, i; = 2.920)

ri

0,’

ci

(nm)

,t);Xin~r)

$rrin-rj

(cm’s_‘)

5.0

7.0 10 60

0.3 0.3 0.3 0.3

3.0 3.0 3.0 3.0

2.18 x 1.12x 5.52 x 1.77 x

low IO_” 1om4 1om5

- 7.41 - 14.4 - 29.2 -911

2.89 x 5.62 x 1.14x 3.55 x

10’ IO’ 103 10”

0.101 0.0722 0.0507 0.00905

3.0 5.0 7.0 10

1.5 1.5 1.5 1.5

15 15 15 15

6.08 x 2.18 x 1.12x 5.52 x

lo- A 1om3 lo-3 lo-”

~ -

5.17 1.43 2.79 5.64

10’ 103 103 lo3

0.075 I 0.0452 0.0323 0.0227

1 .o

I

0.8 -

I

Pe

I

13.3 36.4 71.5 145

I

x x x x

I

I

1

____ ?J= 0.101 -._ 6 = 0.0507 6 = 0.00905

No diffusion

0.6 % 0.4 -

0.2 -

0.0 I 0.0

I 0.2

I

0.4

I

0.6

I

_r

0.8

1.0

1.2

1.4

Z&J Fig. 4. The theoretically

predicted diffusional transfer function for the radial values of the diffusional broadening parameter.

DMA for different

Resolution

of the radial

DMA

1189

shows the diffusional transfer function & for the radial DMA. The nearly triangular curve that closely approximates the ideal, nondiffusional transfer function (dotted corresponds to singly charged particles with diameters of 60nm for the radial operated at the low flow rate of (Qsh = Qe = 3.0 1min-‘, Qa = Qs = 0.3 1min- ‘). lations for 10 and 5 nm particles exhibit more pronounced diffusional distortion radial DMA transfer function.

DIFFUSIONAL

EFFECTS

IN THE

solid curve) DMA Calcuof the

TDMA

The transfer function necessary to deduce the particle size distribution from radial DMA measurements could be determined directly using a truly monodisperse aerosol. In practice, DMAs are usually calibrated using the tandem differential mobility analyzer (TDMA) (Rader and McMurry, 1986; Stolzenburg, 1988). In this technique, two DMAs are operated in series, the first providing a classified aerosol to the second. Because particles transmitted through the first DMA span a finite range of sizes, direct evaluation of Qd is not possible. However, when identical radial DMAs are operated with identical flow rates, validation of the theoretical transfer function is possible when the transfer function of the first DMA is not known a priori. Our focus here is on small particles for which diffusion is significant. Since the probability of such particles acquiring more than one charge in the bipolar neutralizer through which they pass before entering RDMA-1 is small, we can reasonably limit our attention to singly charged particles (Wiedensohler, 1988). The number of singly charged particles of appropriate polarity in a size interval between D, and D, + dD, at the inlet to RDMA-1 is n@,)d4. The transfer function of a DMA, !2(2,, V), is defined as the fraction of particles of mobility Z, that, upon entering the classification region of a radial DMA operated at voltage I/, will exit the DMA through the classified-aerosol outlet port. It is this transmission efficiency that has been modeled above in the derivation of ?&(Z,, V). Any DMA will suffer additional losses in the entrance and exit regions of the instrument. The focus of this paper is on the broadening of the transfer function due to Brownian diffusion, but the experimental results are also influenced by these losses, so they must be treated in the data analysis. If y,,,(D,) is the efficiency of penetration of particles through the entrance and exit regions of the DMA, the total number of particles in this interval at the outlet to RDMA-1 is, (56) The penetration efficiency can be related to the processes that govern losses of particles within the radial DMA. Particle deposition in the radial DMA occurs primarily in the entrance and exit regions of the instrument where both Brownian diffusion and electrophoretic migration transport particles to the channel walls. Diffusional deposition can be expressed in terms of the mass transfer coefficient, k+

(57) a”

where J is the diffusional particle flux, and N,, is the average number concentration in the flow. Although we cannot derive analytical expressions for the complex flows in the entrance and exit regions of the radial DMA, we can deduce scaling relationships by analogy to the results for particle deposition in laminar pipe flow. Friedlander (1977) determined the mass transfer coefficient for laminar pipe flow by applying the so-called Graetz solution wherein the local mass transfer coefficient at a position x along the length of the tube is 113

,

(58)

1190

S.-H. Zhang

and R. C. Flagan

where Pe, = Ud/g is the Peclet number for the pipe penetration efficiency for a pipe of length L of

flow. This

leads

to an overall

(59) The appropriate aspect ratios (L/d) and coefficients for the two regions of interest in the radial DMA are not known. However, since the Peclet numbers for the entrance and exit regions are, for fixed flow rate ratio, proportional to that in the classifier region, it is appropriate to employ the migration Peclet number in the scaling given by equation (59). The penetration efficiency should, thus, scale as ~j~= exp[ - s( Pe-2!3],

(60)

where the coefficient a will be treated as a fitting parameter in the discussion that follows. Particle losses due to electrophoretic migration result from the local electric fields in the exit region of the radial DMA where the aerosol passes from a region of high voltage at the outlet of the counter-electrode to the grounded exit plumbing, as illustrated in Fig. 5. Without analyzing the detailed flow and electric fields, we can deduce the scaling behavior of these losses by noting that, as in the body of the radial DMA, the mean particle motion will follow trajectories of constant particle stream function. The electrical flux function for this junction will again scale with the applied voltage &(r, z) = g(r, z) v.

(61)

The mobilities of the particles being transmitted are centered in a narrow region about Zz, so, for the present purposes, we shall examine the transmission of particles of this mobility. The range of fluid streamlines crossed by the particle is due to the electric flux difference between the center hole of the electrode in the radial DMA and the grounded outlet tubing. Thus, we expect Ic/, - $b = a’z,* I/,

(62)

Grounded

Flow

Out let

Connect ion

Fig. 5. Schematic illustration of electric field lines in the outlet region of the radial DMA that lead to electrophoretic losses of particles as the aerosol exits the classifier.

Resolution of the radial DMA

1191

where /I’ is a geometry parameter for the outlet flow. However, since I/ is inversely proportional to 2; and only particles with mobilities near Zt need be considered, the range of stream functions crossed is a function of geometry and flow rate, but is independent of the applied voltage. Thus, qpen, z should be independent of mobility as well. The penetration efficiency is a product of the efficiency due to diffusional losses and that due to electrophoretic migration, i.e. ‘1Pen

=

qpen,Eexp[-

aPe-2’3].

(63)

In the absence of a detailed flow model for all passages in the radial DMA, the coefficient CIand the electrophoretic penetration efficiency qpen,E must be determined empirically for the complex geometry of the radial DMA. Particles are counted using a condensation nucleus counter that has a counting efficiency of q&D,). Flow through the tubes and valves between the DMA outlet and the CNC also lead to some losses. The transmission efficiency of this plumbing is qp(Dp). The number concentration measured at the outlet to RDMA-1 is

NlWl)

=

(64)

~pen,1(Dp)ndl(Zp(Dp),I/l)il~N~(Dp)~p(Dp)ns(Dp)dDp.

The concentration of particles measured downstream of RDMA-2 is determined by the penetration efficiency and transfer function of the second instrument. For identical instruments operated at the same flow rates, we may assume that in a narrow size range, V) = Qd(Zp(Dp), V). Since the &en,1 = YIpen,* = vpen9 and ndi (Zp(Dp), V) = &(Zp(Dp), same CNC is used for measurements downstream of both RDMA-1 and RDMA-2, and the plumbing is matched for the two measurements to minimize differences in particle losses, the number concentration measured downstream of RDMA-2 is

N2W1,

V2)

m &n.(Dp)~d(Zp(Dp),

=

vl)szd(zp(~p~~

T/2)rcNc(D,)?,(D,)n,(D,)dD,.

s 0

(65)

The influence of Brownian diffusion on the performance employing identical instruments can be seen from the ratio

of the tandem

radial

DMA

(66) The counting and plumbing penetration efficiencies may be combined with the source size distribution into an effective size distribution seen by the radial DMAs, %(Dp)

The measured

concentration N2(V1,

M(Vl,

v*,Ylpen,q

=

~pe”9cNc(Dp)qp(Dp)n,(Dpl

(67)

ratio then becomes V2)

= NIV,)

The dependence of the size distribution of the aerosol entering RDMA-2 on qpen(Dp) is automatically taken into account when n,(L),) is evaluated by the method described below. n,(D,) can be determined once all of the efficiency parameters are known. To calculate Jy; we need to know only qpen(Dp) and n,(Dp). These are determined from the experimental observations by the procedures described below.

1192

S-H.

Zhang

and R. C. Flagan

EXPERIMENTAL

TDMA

system

The diffusional broadening of the radial DMA transfer function was examined experimentally using a tandem differential mobility analyzer (TDMA) system, illustrated in Fig. 6, that employed two identical radial DMAs. The first radial DMA (RDMA-1) was operated at a fixed voltage to generate a monomobility aerosol. The size distribution of that aerosol was measured by stepping the voltage of the second radial DMA (RDMA-2). The resulting measurements are a convolution of the transfer functions of the two identical instruments (Kousaka et al., 1985; Stolzenburg, 1988). Identical radial DMAs were used, and the instruments were operated at identical flow rates. The pressure drop between the two instruments was less than 250 Pa. Thus, the performance of the two instruments should be the same, facilitating study of the transfer function of the radial DMA. The radial DMA characterization experiments were conducted at two different flow rates while maintaining a constant flow rate ratio of p = 0.1. A low flow of Qsh = Q, = 3.01 min-’ was used to probe responses to relatively large particles at a relatively long residence time, while a higher flow of Qsh = Qe = 15 1 min- ’ was used to probe the response to smaller particles. The concentration of particles classified by RDMA-1 at a voltage of I/,, N,(L’,), was measured directly using a condensation nucleus counter (CNC; TSI Model 3025). Alternately, by simultaneously switching two three-way valves, the flow leaving RDMA-1 could be sent through RDMA-2, operating at voltage V,, and then to the same CNC, leading to a concentration measurement N,(V,, I’,). The lengths of tubing, bends, and fittings on the two flow paths were matched to minimize differences in the particle losses exterior to the radial DMAs during the two measurements. The ratio of these two concentrations N, (Vi, I’,)/ N i (Vi), measured at a number of voltages centered about I/, is used to probe the transfer function. Two aerosol sources were employed in these experiments: (i) sodium chloride particles that were used for measurements in the 10-100nm diameter range were generated with

Nitrogen

Vent

I

I

I

?

I

Q i

I RDMA2

Laminar Flow

Filter

Fig. 6. Tandem

radial DMA system used for experimental examination ing of the radial DMA transfer function.

of the diffusional

i

TSI CNC 3025

broaden-

Resolution

of the radial

DMA

1193

a constant rate nebulizer; and (ii) ultrafine platinum particles that were used for measurements in the 3-10nm diameter range were produced using a repeating spark source generator similar to that described by Schwyn et al. (1988). The effective size distribution of the source aerosol is well represented by a log normal distribution, n,(ln Dp) d In D, = g:nn,

exp[

- ~(‘“(~r~)~]dln~~,

(69)

where N, is the total number concentration of singly charged particles of appropriate polarity, D,, is the geometric mean diameter, and og is the geometric standard deviation. These three parameters were determined by measuring the concentration Nii at a number Of voltages Vri, i = 1, . . ,m, and then minimizing the functional X2(N,, D,,, 0.J = f

Cln Nij - ln

Nl,(vli,

N,,

Dpg,

~s)12,

(70)

i=l

where Ni, is the total number of particles determined by substitution of equations (67) and (69) into (64). The transfer function Rd(Z,, V) was used in the numerical evaluation of the integral. x2 was minimized with respect to the three parameters using Powell’s method (Press et al., 1992).

RESULTS

The aerosol produced by the spark source typically yielded N, = 2.13 x 10’ cmm3, with Dpp = 8.62nm, and gg = 1.47. As indicated by equation (67) the effective source size distribution is that of charged particles reduced by the penetration, plumbing and counting efficiencies. For the NaCl aerosol source, typical parameters are N, = 1.54 x lo5 cmm3, Dpp = 32.9 nm, and gg = 1.87, respectively. The drift in the total number concentration was estimated to be less than 5% during each run. Figure 7 shows the measured concentration ratios JV( Vi, V,) as a function of V2 for several values of Vi and for two different operating flow rates. The low flow rate data (Fig. 7(a), Qa = 0.3 1min- ‘) for 60 nm particles have a sharp, symmetric peak, with a peak value OfJV pk = 0.49. The peaks for smaller particle sizes (indicated as centroid mobility diameters for RDMA-1) are lower and broader. The peak concentration ratio for the 5 nm particles is only JV& = 0.26. The measurements shown in Fig. 7(b) for a higher flow rate of Qa = 1.5 1min-’ exhibit a similar trend, with 3 nm particles producing a peak value of J$ = 0.28. The theoretically predicted concentration ratios are compared with the experimental measurements in Fig. 8. The predictions based on the nondiffusional transfer function, shown by the dashed line, are much more sharply peaked than the experimental data. The diffusional transfer function reproduces the general shape of the observed peak, although the experimental data peak at a slightly lower mobility (larger particle size) and exhibit a lower peak concentration ratio than predicted. The height of the peak in JV can readily be explained by diffusional losses of particles to surfaces in the DMA. The apparent shift in the location of the transfer function peak for small particles has previously been observed in TDMA experiments conducted using cylindrical DMAs (see Kousaka et al., 1985; Stolzenburg, 1988, and references therein). No satisfactory explanation for this apparent growth of fine particles between the two DMAs has been put forth. In order to fit his theoretical predictions to his data, Stolzenburg (1988) introduced a correction to the voltage of the second DMA to offset this discrepancy. For the data shown in Fig. 8, analyzing the data as if the operating voltage of RDMA-2 were lower than the value supplied by a factorf, = 0.96 yields much closer agreement between the calculated and observed particle size distributions in the ultrafine particle TDMA experiments. The peak in the TDMA measurements is somewhat broader than the predicted distribution for creeping flow in the radial DMA. Again, following Stolzenburg (1988), we increase 5 by a factory, = 1.07 to match the shape of the TDMA peak.

1194

S.-H. Zhang

and R. C. Flagan

(W

0

3nm

W

5nm

*

7nm

3

?? 10nm 1

Fig. 7. Ratio of the outlet to inlet concentrations for RDMA-2 as a function of - I/,: (a) low flow rates, Q.,, = 3.0 1min- I, (b) high flow rates, Q+, = 15 1min- I.

/\

2”

= 0.3

I min.' I min.'

!/’ 11 I,/ ’’ +7nm 1 A Data ,/ ’ 3I 1 ---- ideal , 1 --_-Q

0.8-

t

$

Qh = 3.0

0.4-

0.2-

0.0-n-A-’ 0.6

0.8

1.0 &MA2

1.2

1.4

/Zp,RDMAI

Fig. 8. TDMA concentration ratios for 7 nm nominal diameter at flow rate Qsh = 3.0 Imin-’ as a function of normalized particle mobility showing raw data (points), predictions without correction factors (dotted curve), and predictions made using correction factors that minimize the weighted sum of the squared deviations between theory and experiment (solidcurve).

Thus, we have used Stolzenburg’s (1988) model to predict the form of the transfer function, and used empirically determined correction factors to match the location and shape of the TDMA distribution. The physical significance of the penetration efficiency, pen, is obvious, but that off” and f6 is not. Because both deviations have been observed for 9 radically different DMAs, one may surmise that some, as yet unexplained, mechanism is responsible for the deviations. Following Stolzenburg (1988), we present the latter two correction factors, hoping that physical interpretation will follow. The parameters were evaluated by minimizing the weighted sum of squared deviations for each I/, ,

(71)

Resolution

of the radial

DMA

1195

using Powell’s method (Press et al., 1992). Since measurements were made over a much wider range of mobilities than are important in the transfer function, the fitting was performed over all data for which Jf exceeded 5% of the peak value to avoid placing undue emphasis on the tails of the transfer function. N,j/‘Nr is the concentration ratio measured at l/,j. N(V, ,fv V'zj, qpen, f6)is the model prediction using equation (68), a log normal inlet distribution, equation (69), and the correction factors as described above. The uncertainty in the weighting factor 0,: , was based on the counting statistics and error propagation in the determination of the concentration ratio, i.e. (72) which leads to (73) Table 2 presents average values of the fitting parameters determined by this procedure for a number of particle sizes and two different flow rates and a flow rate ratio of fi = 0.1. TDMA voltage corrections are smaller than 8%. Greater variability is seen in the values of fa. The large value for this parameter for large sizes may be indicative of some distortion of the flow field near the entrance slit that distributes the particles over a wider region of the flow than would ideally occur. This effect would be larger for more sharply peaked data. Figure 9 compares the measured concentration ratio N( VI, V2) as a function of Zp*JZp*r with predictions made using these correction factors for several values of Zt, and for the two different operating flow rates. The estimated values of lognpen are plotted versus Pe-2’3 in Fig. 10. The penetration efficiencies determined from the TDMA measurements follow the expected dependence on the Peclet number, with a best fit (R2 = 0.86) of ripen = 0.98 exp [ - 5.6 Pe-2’3].

(74)

The inferred penetration efficiencies are higher than those that we previously reported for the radial DMA. The previous estimates, which were based upon the ideal, nondiffusional transmission function, overestimated the particle losses in the radial DMA. This discrepancy clearly demonstrates that the operating characteristics of a DMA must be based upon the true transfer function of the instrument. The high quality fit of the TDMA data obtained using the Stolzenburg (1988) diffusional transfer function strongly supports the present analysis, although the nonunity fzvalues shown in Table 2 suggest that there may still be room for improvement. The voltages on the RDMA-2 forthe calculated response functions that are presented in Fig. 9 were adjusted by the factor fv.The apparent mobility shift was found to depend on

Table

2. Averaged

gm,

Aerosol

5 7 10 20 40 60 3 5 7 10 20 40 60

Pt Pt Pt NaCl NaCl NaCl Pt Pt Pt Pt NaCl NaCl NaCl

parameters

determined

by fitting TDMA predictions

data

to diffusionally

broadened

c&-3) 4180 9610 10300 3690 4960 3380 301 4410 10200 10900 --393Q 5280 3570

3 3 3 3 3 3 15 15 15 15 15 15 15

-

7.41 14.4 29.2 114 429 910 13.3 36.4 71.5 145 563 2120 4510

0.872 0.876 0.946 0.977 0.988 0.995 0.918 0.907 0.949 0.962 0.978 0.975 0.984

0.934 0.950 0.98 1 0.968 0.970 0.975 0.979 0.984 1.01 0.997 1.011 0.998 0.990

1.04 1.11 1.27 1.55 2.76 4.23 1.29 1.36 2.01 2.49 4.53 7.10 13.6

0.101 0.0722 0.0507 0.0255 0.0131 0.00905 0.0751 0.0452 0.0323 0.0227 0.0115 0.00593 0.00407

0.104 0.0799 0.0644 0.0395 0.0363 0.0383 0.0969 0.0615 0.0648 0.0566 0.0521 0.0421 0.0555

transfer

function

wth

a srp

3.89 5.08 6.39 8.23 9.04 9.32 4.93 6.78 7.74 8.43 9.16 9.55 9.68

3.75 4.77 5.45 7.16 7.44 7.93 4.08 5.61 5.52 5.99 6.29 7.21 6.06

S.-H. Zhang

1196

I

0.41

I

’

and R. C. Flagan

I

I

(A) q = 0.3 I min.’ qh = 3.0 I min.’

f 9

5 nm A ____ 5 nm fit ?? 7 nm _.._. 7 nm fit 10 nm 0 10 nm fit

0.2-

0.1 -

0.1

-

J

0.8

1.2

0.0

0.4

1.2

0.8

i p,RDMAdZ’p,RoMAI

i p,RDMA2/Z’p.RoMAI

Fig. 9. Comparison of fitted TDMA response function with experimental observations for 5,7, and 10 nm diameters at (a) low flow rate, Qsh = 3.0 I min- ‘, and (b) high flow rate, Qsh = 15 1min- ‘.

lqs

.. A

I l I-----&-L-__

I

9-

I

.A

A 0-

711 pen

65-

V

open fit QJQ,, = 1915,

A

C&/C&

= 1.5/15,

0

Q,/Q,,

= 0.3/3,

W

Q$Q,,

= 0.313, spark

I

co

NaCl spark NaCl

I

0.5

I

I

I

1.0

1.5

2.0

2.5x10-2

-213

Fe Fig.

10. Measured

penetration

efficiency

for the radial DMA as a function Peclet number.

of particle

migration

both the particle size and on the flow rate through the radial DMA. As shown in Fig. 11, the factorf, was found to correlate reasonably well with the particle migration Peclet number, and was fitted by the expression fv = 0.99 - E

(75)

with R2 = 0.77. The values off” shown in Fig. 11 appear to be smaller at low flow rates than at higher ones. Stolzenburg (1988) suggests that observed values of fV < 1 might be explained by a decrease in the particle mobility within the TDMA system, i.e. an increase in the particle size, although no satisfactory mechanism for this change has yet been reported. The causes of this shift will require further investigation. The significance of the enhanced broadening of the transfer function that is reflected in f6 is best seen by examining the resolution %. The resolution has been calculated assuming that the shape of the transfer function is that predicted using Stolzenburg’s (1988) model with the predicted value of 6 increased by the factor&. Figure 12 compares the observed

Resolution

of the radial

DMA

0

1197

Q$Q,

= 0.313

?? c&/Q, -

= 1.5/15

f, fit

1.00

f”

0.9

o.88 t

0

1

2

3

4x1 os

w-l Fig.

11. Variation

of the voltage

correction factor for RDMAZ migration Peclet number.

20

I -

1 ’ ’ ’ ““I

the TDMA

system with particle

’ ’ n ’ 1’1’1

Theory: RDMA A

RDMA: 0,

= 3

I min”

RWA: Cl, = 15 I min.’ - - Theory: CDMA, p=O. 1 1 5 _ A CDMA: bO.1 - - - Theory: CDMA, p=O.OS V CDMA: fkO.05

.’

v

, ’ N’

I

, .*’

v

v

,’ .‘V

,’

O”..“‘.’

Fe Fig. 12. Predicted and observed DMA resolution as a function of particle migration Peclet number for the radial DMA (solid points) and for the Knutson and Whitby cylindrical DMA (open points; data of Stolzenburg, 1988).

and predicted values of L%as a function of the migration Peclet number for the two flow rates at which the radial DMA was operated. The measured resolution falls below the predicted value, apparently approaching a lower asymptotic value than is predicted. Moreover, the deviation is higher for the high flow rate than for the lower one. For comparison, thef, values estimated by Stolzenburg (1988) for the Knutson and Whitby

1198

S.-H. Zhang and R. C. Flagan

cylindrical DMA have been used to estimate the resolution for that instrument. purpose, the migration Peclet number in the cylindrical DMA is defined as

PeCDMA

=

(Qsh + QeI(R2 - RI) h!.R2%l

’

For this

(76)

where R2 and RI are the outer and inner radii of the classifier column, respectively, and L is the column length. Two different flow rate ratios were used in the cylindrical DMA experiments, /I = 0.1 and 0.05. Measurements for these two conditions are shown as open triangles in Fig. 12. The cylindrical DMA data more closely follow the theoretical predictions than do the radial DMA data. For p = 0.05, however, the measurements deviate from the predictions for Pe > lo3 and appear to tend toward a lower asymptotic limit than the simple theory would suggest. The resolution of both the radial DMA and the cylindrical DMA appear to be degraded by fluid mechanical effects, possibly the icsult of flow disturbances near the entrance slot or at the classified aerosol exit. There are two differences between the flows in the radial DMA and the Knutson and Whitby cylindrical DMA that might explain the larger degradation that is observed in the radial DMA. The aerosol is introduced into the radial DMA through a tangential flow that produces higher shear than is encountered in the Knutson and Whitby cylindrical DMA. This shear may lead to flow instabilities that could distribute the incoming aerosol more broadly than the simple models assume. Thus, although the tangential inlet significantly enhances particle penetration in the radial DMA over that through the long narrow annular passage in the Knutson and Whitby DMA, it may be responsible for reduced resolution. This could be tested by examining the performance of the University of Vienna cylindrical DMA (Winklmayr et al., 1991; Reischl, 1991) that also employs a tangential inlet. The other possible cause of degraded resolution is the high converging gas velocity at the center of the radial DMA. Flow instabilities or turbulence at the sample extraction point could again distort the instrument response. Since the high mobility particles are deposited on the electrode containing the sample extraction port on the radial DMA, mixing at the outlet port would tend to bias the measurements toward particles of lower mobility. This is the direction of the mobility shift shown in Fig. 9. However, since this mechanism is unlikely to be important in the cylindrical DMA, the voltage shifts observed in the cylindrical DMA by previous workers must be explained before this can be taken as indicating flow distortions associated with the central outlet. Again, a comparison of the resolution with another instrument would help resolve this contribution. The radial DMAs described by Mesbah (1994) include a version in which the aerosol is extracted through an annular slot rather than a central outlet port. That instrument would not involve the high radial velocities in the classified aerosol flow that would be required for this instability mechanism, and could, therefore, provide a test of this mechanism.

CONCLUSIONS

The model of particle migration with diffusion that Stolzenburg (1988) first derived for the cylindrical DMA has been extended to the radial DMA and compared with experimental observations of the particle size distributions measured in TDMA experiments using stable aerosols. The diffusional transfer function model reproduces the trends of the experimental observations for particles as small as 3 nm diameter, although small adjustments in the transfer function parameters are required to bring the predictions into quantitative agreement with the experimental observations. Diffusional broadening of the transfer function is pronounced for these small particles. Particle losses in the radial DMA are low and well correlated with the particle Peclet number. Indeed, the penetration efficiencies determined in the present study are higher than we estimated in our initial description of the radial DMA (Zhang et al., 1995). This difference results from analysis of the instrument using the diffusion-broadened transfer

Resolution

of the radial

DMA

1199

function derived by Stolzenburg (1988) rather than the ideal, non-diffusional transfer function that we used in our earlier paper. Thus, diffusional broadening effects must be taken into account when evaluating the performance of DMAs in general. As in previous studies of cylindrical differential mobility analyzers, an apparent shift in the particle mobility between the two DMAs of the TDMA system has been observed with the radial DMAs. Although this shift has not been explained, it has also been found to correlate well with the particle migration Peclet number. When coupled with the model for the diffusional broadening of the transfer function, the experimental correlations provide a basis for inverting radial DMA measurements of ultrafine aerosol particles. The model reproduces the shape of the size distributions measured in TDMA experiments quite well, although the model does predict a slightly higher resolution than is observed. A similar, but smaller degradation in the resolution is observed in Stolzenburg’s (1988) cylindrical DMA data. The degradation appears to result from flow non-idealities in the radial DMA, although the origin of those non-idealities remains unclear. A detailed and self-consistent comparison of the performance of DMAs of different designs should be very helpful in identifying the causes of degraded performance, and ultimately, to the development of instruments that share the benefits of high penetration efficiency and high resolution.

Acknowledgments-The authors wish to thank Dr Mark Stolzenburg for important suggestions regarding the treatment of the diffusional transfer function. This work was supported by the National Science Foundation grant NSF-ATM-9307603, the Office of Naval Research Grant NOOO14-93-1-0872, and the Environmental Protection Agency Exploratory Environmental Research Center on Airborne Organics (R-819714-01-0).

REFERENCES Bird, R. B., Stewart, W. E. and Lightfoot, E. N. (1960) Transport Phenomena, p. 114. Wiley, New York. Fissan, H., Hummes, D., Stratmann, F., Neumann, S., Pui, D. Y. H. and Chen, D. (1995) Comparison of four differential mobility analyzers for nanometer aerosol measurements: transfer function evaluation. In Proc. Sixth European Symp. on Particle Characterization. Niirnberg Messe GmbH, Niirnberg, pp. 391400. Fissan, H., Hummes, D., Stratmann, F., Biischer, P., Neumann, S., Pui, D. Y. H. and Chen, D. (1996) Experimental comparison of four differential mobility analyzers for nanometer aerosol measurements. Aerosol Sci. Technol. 24, 1-13. Friedlander, S. K. (1977) Smoke, Dust and Haze, p. 77. Wiley, New York. Fuchs, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. Knutson, E. 0. and Whitby, K. T. (1975a) Aerosol classification by electric mobility: apparatus, theory, and applications. J. Aerosol Sci. 6, 443-451. Knutson, E. 0. and Whitby, K. T. (1975b) Accurate measurement of aerosol electric mobility moments. J. Aerosol Sci. 6, 453460. Kousaka, Y., Okuyama, K. and Adachi, M. (1985) Determination of particle size distribution of ultra-fine aerosols using a differential mobility analyzer. Aerosol Sci. Techno/. 4, 209-225. Kousaka, Y., Okuyama, K., Adachi, M. and Mimura, T. (1986) Effect of Brownian diffusion on electrical classification of ultrafine aerosol particles in differential mobility analyzer. J. Chem. Eng. Japan 19, 401407. Liu, B. Y. H. and Pui D. Y. H. (1974) A submicron aerosol standard and the primary, absolute calibration of the condensation nuclei counter. J. CoUoid Interface Sci. 47, 155-171. Mesbah, B. (1994) Le Spectrometre de Mobilite Electrique Circulaire: Theory, Performances et Applications. Ph.D. Thesis, University of Paris. Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992) Numerical Recipes in Fortran. Cambridge University Press, Cambridge. Rader, D. J. and McMurry, P. H. (1986) Application of the tandem differential mobility analyzer to studies of droplet growth or evaporation. J. Aerosol Sci. 17, 771-787. Reineking, A. and Porstendiirfer, J. (1986) Measurements of particle loss functions in a differential mobility analyzer (TSI Model 3071) for different flow rates. Aerosol Sci. Technol. 5, 483486. Reischl, G. P. (1991) The relationship of input and output aerosol characteristics for an ideal differential mobility analyzer particle standard. J. Aerosol Sci. 22, 297-312. Rossell-Lompart, J. and de la Mora, J. F. (1993) Minimization of the diffusive broadening of ultrafine particles in differential mobility analyzers. In Synthesis and Measurement of Ultrafine Particle (Edited by Marijnissen, J. C. M. and Pratsinis, S. E.), pp. 109-l 14. Delft University Press, Delft. Russell, L. M., Stolzenburg, M. R., Zhang, S. H., Caldow, R., Flagan, R. C. and Seinfeld, J. H. (1996) Radiallyclassified aerosol measurements. J. Atmospheric Oceanic Technol. 23, 598-609. Schwyn, S., Garwin, E. and Schmidt-Ott, A. (1988) Aerosol generation by spark discharge. J. Aerosol Sci. 19, 639-642. Stolzenburg, M. R. (1988) An ultrafine aerosol size distribution measuring system. Ph.D. Thesis, University of Minnesota.

1200

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and R. C. Flagan

Stratmann, F., Hummes, D., Kauffeldt, Th. and F&an, H. (1995) Convolution and its application to DMA transfer function measurements. J. Aerosol Sci. 26, S1433S144. Wiedensohler, A. (1988) An approximation of the bipolar charge distribution for particles in the sub-micron size range. J. Aerosol Sci. 19, 387-389. Winklmayr, W., Reischl, G. P., Lidner, A. 0. and Berner, A. (1991) A new electromobility spectrometer for the measurement of aerosol size distributions in the size range from 1 to 1000nm. .I. Aerosol Sci. 22, 2899296. Zhang, S. H., Akutsu, Y., Russell, L. M., Flagan, R. C. and Seinfeld, J. H. (1995) Radial differential mobility analyzer. Aerosol Sci. Technol. 23, 357-372.