Inversion of aerosol data from the epiphaniometer

J Aerosol Sct, Vol 22, No 4 pp 417 428 1991

0021 8502'9l $ 3 0 0 + 0 0 0 ( , 1991 Pergamon Press pie

Printed an Great Britain

INVERSION OF AEROSOL DATA FROM THE EPIPHANIOMETER SPYROS N. PANDIS, URS BALTENSPERGER,* J. KENNETH WOLFENBARGER a n d JOHN H. SEINFELD Department of Chemical Engineering, California InsUtute of Technology, Pasadena, CA 91125, U S A (Recewed 20 August 1990, and m Jmal form 29 January 1991) Abstract--The epxphanlometeris a new instrument for continuous monltormg of the Fuchs surface of aerosol particles using the attachment of neutral radloactl,,e 2t 1Pb atoms to the particles Sincethe 2XlPb atoms have a half-hfe of 36 1 mm. the ablht~ of the eplphamorneter to detect short-term fluctuations m the Fuchs surface of the aerosol is hmlted To overcome this drawback a general algorithm developed for the inversion of size dxstnbut~ondata from conventional aerosol instruments is adapted for the mverslon of data from the eplphanlometer This algorithm allows resolution of dramatic changes in the aerosol Fuchs surface occurring over a txmescale of minutes The method IS applied to data collected during outdoor smog chamber experiments, with changes in aerosol concentration of several orders of magnitude occurring within a time period of 5 mm The inverted eplphamometer data are in agreement ~lth the corresponding data from a Scanning Electrical Mobility Spectrometer (SEMS) and an Optical Particle Counter

INTRODUCTION Recently the epaphaniometer (Greek eptphama, the surface of a body) was m t r o d u c e d as a new i n s t r u m e n t for c o n t i n u o u s aerosol m o n i t o r i n g (G~iggeler et al., 1989). The devxce determines the F u c h s surface of aerosol particles, using the a t t a c h m e n t of n e u t r a l radioactive 2 1 i p b a t o m s to the aerosol particles The e p i p h a n i o m e t e r has been used for tn sttu d e t e r m i n a t i o n of the exposed surface of agglomerated aerosol particles (Schmidt-Ott et al., 1991), for the m e a s u r e m e n t of the mass transfer to spherical a n d agglomerate aerosol particles (Rogak et al, 1991), a n d for c o n t m u o u s m o n i t o r i n g of a m b i e n t aerosols at highalpine sites (Baltensperger et al., 1991). Since the 211Pb a t o m s attaching to the aerosol particles have a half-life of 36 1 mln, the ability of the e p l p h a n l o m e t e r to detect short-term fluctuations m the Fuchs surface of an aerosol is limited. Therefore the use of the i n s t r u m e n t alone for the study of processes with time scales shorter t h a n a n hour or so was n o t possible C o m p u t a t i o n a l data inversion m e t h o d s can, however, be used to overcome this drawback. In this work a m e t h o d is developed for the c o n t i n u o u s calculation of the Fuchs surface of an aerosol d i s t r i b u t i o n as m e a s u r e d by the e p l p h a m o m e t e r even in the case of variations in the aerosol occurrmg over a time scale of m m u t e s This m e t h o d extends considerably the range of the potential applications of the eplphantometer. We begin by p r e s e n t m g a short description of the e p i p h a n i o m e t e r i n c l u d i n g e v a l u a t i o n of its residence t~me d i s t r i b u t i o n (RTD) a n d its response function. I n the next section a rigorous defimtlon of the F u c h s surface is presented together with a m e t h o d for its calculation for a n y spherical particle The mversion algorithm is described next a n d is evaluated for several test problems. Finally, the inverted data from the e p i p h a n t o m e t e r are c o m p a r e d w~th those collected by c o m b i n i n g a S c a n n i n g Electrical Mobility Spectrometer (SEMS) a n d an Optical Particle C o u n t e r (OPC) d u r i n g o u t d o o r smog c h a m b e r experiments with changes in aerosol c o n c e n t r a t i o n of three orders of m a g n i t u d e occurring within a time period of a few minutes.

* On leave from Paul Scherrer Institute, CH-5232 Vllllgen,Switzerland 417

I-HF F P I P H A N I O M I ~ T F R

A detailed descrlpDon of the eplphamometer Js given by Ghggeler e! a/ (1989i. :\n a2~ stream containing the particles is pumped Inside the eplphamometer to a chamber containing a 22"Ac source By consecutive radioactive decay 227Ac produces 2O,pb ~~a the following decay chain (Seelmann-Eggeber¢ et al. 1981~' 227Ac(T1,2 = 2 t 8 y) (fl)-*":~Th( 7 ~ : = 18.7 d) (0~)-*2Z3Ra(Tj ,,2 =- t 1.4 d) (~)--~

z I o Rn ( 7"1 ~ = 3.96 s) (~)--~-~15 Po ( T: . = 1 8 ms) (~)--*211 Pb ( T, , = ~0 1 mm) (/¢~-~ 21tBl( T~ 2 = 2 17 ram)(~)_~2o~ Tlt I-~ .. = 4.77 mm)t/~}~z°TPb with (fl) and (~) indicating decay by fl- and ~-emlsslon Due to its short half-life, 219Rn emanating from the solid source diffuses over a distance of only a few cm before it decays further to 211pb. The 21:pb (and 211Bi) atoms attach to aerosol particles pumped through the main epiphaniometer chamber. After a residence time of around 2 min in the chamber (for a typical gas flow rate of 1 1rain- ~), the aerosol particles are transported through a capillary that acts as a diffusion barrier for non-attached lead atoms. At the end of the capillary the particles are deposited onto a filter. The activity of 21~pb deposited on the filter is measured continuously by an annular surface barrier detector via the ~-act~vity of its daughter 2tlBi. Epiphaniometer response func turn

The epiphanlometer signal (number of the measured alpha decays) depends on the residence time of 211pb (or 211Bi) and aerosol in the vessel, the attachment coefficient and the detector efficiency If one connects an aerosol source to an epiphanlometer that has a clean filter, then 0t-decay is observed approximately 2 min after the aerosol first enters the vessel. This fact suggested that one can approximate the lead and aerosol residence time distribution (RTD) as one of plug flow. The validity of this assumption will be examined later. If a particle population has an inmal number n o of 211pb atoms and no 211Bi attached to it, then the number of 211Bi atoms on the particle population at time t, n2[t) is given by (Friedlander et al, 1981) ~--~e

h e ( t ) = -: Z2

-

- e

(1)

L t

with 21 and 2z the decay rate constants of 211pb and 2tlBi correspondingly given by In 2 21 - T1/2 [zl 1pb ) = 0.0192 mm -~,

In 2 ~'2-- Tl,,(ZllBi) =0-3194 min-1

12)

If one assumes that the aerosol flow starts at t = 0, the filter is initially clean, and that the plug flow assumption holds, then the concentration of ZllBi on the filter, N z (t) is found directly from equaUon (1): N2(t)=0

Nz(U=

t<_r

21

22--21

Ii -~

jo

q(s)(e-~l~'-~)-e ~c~-~))ds

t>r,

(3)

where z is the residence time in the vessel, and q(t) is the rate at which 211Pb atoms (including their daughters) reach the filter at time t + z. The epiphaniometer signal or equivalently the number of counts measured in the time interval [t,_ 1, t,] is y, = w£z

Nz (t)dt,

where w is the counting efficiency of the detector.

(4)

Aerosol data from the eplphamometer

419

At this point one can check the validity of the plug flow assumption comparing the instrument response to a step function input with that predicted by equation (4) that uses this assumption. If at time t = 0 a steady aerosol input is connected to the epiphaniometer then equation (3) can be integrated to give 21 I e - a ' ~ - - e -~lt N~(t)= 22 ~ 21 q 2i

e - ~ - - e -~2t] )~2 '

(5)

Introducing the expression (5) into equation (4) one obtains the theoretical response of the epiphaniometer to a step function input, -

-

_

212

-

22

.

(6)

Experimentally (Fig. 1) a steady aerosol input was produced and connected to the epiphaniometer inlet at time t = 0 (Weber, 1990). The number concentration of the aerosol was measured at the same time by a Condensation Nuclei Counter (CNC). The response of the epiphaniometer shown by the data prints in Fig. 1 agrees very well with the prediction from equation (6) as shown by the 'theory' line in Fig. 1 validating the plug flow assumption as well as the above theoretical description of the epiphaniometer. In this experiment the integration interval [tl- i, t,] was 1 rain and the unknown constant q in equation (6) was used as a free parameter. The detector efficiency, w, is given by the solid angle subtended by the detector, since a-particles incident on the detector used are counted with 100% efficiency. For the geometry used in the current version of the epiphaniometer the average efficiency has been calculated to be 20%. Generally, the aerosol concentration is not constant and therefore q = q (t). With a change of variable equation (4) can be rewritten as

y,=f~q(t)k,(t)dt,

(7)

where k,(t) is referred to as the instrument kernel function, is equal to the instrument response to an impulse input and is given by 22

k ' ( t ) = z2 - z i

(e ~ a t - 1)e -~w' - ' ) -

k, (t)= 22 )~2_21 (eal~_ e- zw, -,))

21

22 - Xi

(e a~a'- 1)e -z~"' -'~

2z 2a). - , 1(ca2,_ ) ) _e- a.(:,

t<_ t,_ 1 - z

t,_t_r
k~(t)=0 t , - r < _ t ,

(8)

where At=t,-t,_x is independent of i. Selected epiphaniometer kernel functions for At = 5 min and z = 2 min are depicted in Fig. 2. It is of interest to note that the fraction of 21 i Bi that decays before reaching the filter, f~, increases as z increases: f~ =

22(1 - ea'~)- 21 (1 - e a2,)

(9)

For z = 2 min, less than 1% of the signal is lost by radioactive decay. FUCHSSURFACE Calibration experiments (Gfiggeler et al., 1989) showed that the attachment coefficient of the 211pb and 211Bi atoms can be described by the Fuchs' coagulation theory (Fuchs, 1964; see also Selnfeld, 1986). According to this theory, at small aerodynamic diameters (d< 100 nm), the epiphamometer signal is roughly proportional to the surface area of the aerosol particles This result also holds for agglomerates of non-spherical shape where the AS 22.4-B

420

g

....

q "-I ~

N

Pa~t:,is et
Theory CNC c o u n t

~tso~

- 3 ~(10 em-)

,~"-I

/

~

//

80-1

: !

'

i •~

~-~

~.~,,~_ _~a~i,~,_ _ ~

____,,~

.,,

o !........ ! ~---"- . . . . . . . . . -10

0

10

20 ,30 Time (rnin)

,

i

40

50

Fig. 1 Comparison of the theoretical [plug flow assumpuon, equation (6)] to the experimental response of the epiphamometer to a step function aerosol input. The aerosol number concentration measured by a CNC is also presented

0.10

~ 0.08

!

k40

kso

k60 i

~ 0.08

:00,4/// Z I /I ,!'/ 0.00

-60

,

, 0

, 60

,

li

T 120

Time

1130

240

aoo

(rain)

Fig. 2 Kernel functions for the eplphanlometer for measurements at 50 mln (klo), 100 mm {k2o), 150 min (k3o), 200 min (k,o), 250 mm (k~o) and 300 m l n (k6o) at a counting period of 5 mln

a p p r o p r i a t e length scale is the m o b i l i t y d i a m e t e r ( S c h m i d t - O t t et al., 1991). O n the o t h e r h a n d , at large a e r o d y n a m i c d i a m e t e r s (d > 3/~m) the signal is p r o p o r t i o n a l to d. If N m o n o d i s p e r s e spherical particles of d i a m e t e r d enter the e p i p h a n i o m e t e r then the e p i p h a n i o m e t e r signal s will be s=CI(T

, Cpb ,

W,

.

. ) N K I 2 ( d ),

(10)

where the C1 d o e s n o t d e p e n d on d b u t o n i n s t r u m e n t p a r a m e t e r s like the residence time in the c h a m b e r , the c o n c e n t r a t i o n of the P b a t o m s , the d e t e c t o r efficiency, etc. The sagnal is a linear function of the c o a g u l a t i o n coefficient K 12 between the a e r o s o l particles a n d the lead atoms.

Aerosol data from the eplphamometer

421

To facilitate the interpretation of the epiphaniometer results, we define the Fuchs surface, SF of a particle of diameter d (in/am) as

[ d ~ x(d, sF=~\doj ,

(ll)

where d o = 1/am and x(d) varies between 1 and 2 and will be calculated next. The Fuchs surface defined in this way is a dimensionless quantity. Using this definition the signal of the eplphanlometer will be analogous to the Fuchs surface and can be written as

1

{ d \x(d)

s = ~ - 2 N rtlkfoo)

,

where C2 is a constant. Combining equations (10) and (12) an expression for obtained x(d)=

ln(C K12(d))-ln ~, In(d/do)

(12)

x(d) can be (13)

with C = C1 C2 being another constant. The constant C can be chosen so that x does not tend to infinity as the diameter d approaches 1/am. Then one chooses g

C -- - -

Kaa(do)

(14)

finally obtaining

x(d)=

l n K 1 2 ( d ) - l n K12(do) l n ( d ) - l n (do)

(15)

The last step in the calculation of x is the evaluation of the coagulation coefficient. This evaluation, using the Fuchs form of the coagulation coefficient, requires knowledge of the diameter and the density of the lead atoms attaching to the aerosol. These physical properties are not well known. The lead atoms are probably surrounded by water or other molecules increasing the lead's diameter and decreasing its density. Previous measurements suggest that the effective diameter of the lead cluster is between 0.6 and 2 nm (Suet al., 1988; Porstend6rfer et al., 1979). We assumed first that the lead particles are spherical containing only one lead atom and several water molecules forcing their density to be a simple function of the diameter. To determine their diameter a series of additional experiments was used. In these experiments a known number concentration (measured by a CNC) of monodlsperse aerosol particles of diameter d was connected to the epiphanlometer and the instrument response was determined in order to measure mass transfer to aerosol particles (Rogak et al., 1991) For a given choice of diameter of the lead clusters a unique x (d) can be determined using equation (15) (Fig. 3). The accuracy of this choice can be evaluated by plotting the experimentally measured calibration factor C2 = Nrc(d/do)X(a)/s as a function of d. If our chome was correct, the diagram should be a horizontal line equal to C2 for every d. The results for some of our attempts are depicted in Fig. 4. The best choice of lead particle diameter according to the available information is d = 1.5 nm and the corresponding x (d) curve will be used throughout this work for the calculation of the Fuchs surface of a known aerosol size distribution. At high particle concentrations the eplphaniometer response is no longer linear. Saturation occurs when all the lead clusters produced in the exposure chamber attach to the aerosol particles. When saturated, the measured activity is Smax = 330 Hz regardless of the partmle concentration. The epiphaniometer response becomes non-linear at about 45% of this saturation activity or at around 150 Hz (corresponding to 106 0.01/am particles c m - 3 or 600 1 #m particles cm-3). The correction factor F for these conditions was determined experimentally (Weber, 1990) as F = 1 +0.144g-0.0108g 2 - 4 . 8 8 g 3 + 10.22g*,

422

'% ~)ANDIS et

(d

2 . 0 -~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

dl=O.5 nrn

1.8

"'.,dl=l.O

nm

1.4 -i

d~=2.

i J

"

"'"

1

l.O

i

. . . . . . . .

I0

i

,

' .......

I00

I0~00

Diameter (nm) Fig 3. Exponent x [equattons (11) and (13)] as a funcuon of the Input aerosol diameter d (nm) for four different lead cluster diameters

24 ooooo

d~b

=

1.5

nrn

- .-

dpb = 0.5 nm

......

dpb

20 ¸

=

dpb =

1.0

nm

2.0

nm

NI6.

o ~,.

o

o

c

.o

% [] . . . . . . . . .

oil

n°Q. .....

. -

-~_ 8. O

o

4.

O

0

z60

460

660

a00

O (nm) Fig 4. Calibration factor C 2 as a function of the diameter of the monodlsperse aerosol for a lead cluster diameter equal to 1 5 n m (horizontal sohd line). Also shown (dashed lines) the best fits for diameters 0 5, 1 0 and 2.0 n m (the corresponding data points are not shown). The best choice of lead diameter is the one for which the calibration constant is the same for all diameters

where g = s/sm~x is the measured actwity normalized by the saturation value. The data were corrected with this factor. Summarizing, we defined for every particle its Fuchs surface which can be viewed as an intermediate m o m e n t between its diameter (first moment) and its surface (second moment). By definition, the epiphaniometer signal for a particle is analogous (within a calibration constant) to the Fuchs surface of the particle. Generally the epiphaniometer data alone are

Aerosol data from the eplphamometer

423

not sufficient for the calculation of other moments like number, surface or volume, m the same way that a Condensation Nuclei Counter can provide only the number concentratmn and not any other moments. THE INVERSION ALGORITHM The epiphanlometer can provide a substantial amount of information about fluctuations in the aerosol Fuchs surface with respect to time. However, the instrument signal can be difficult to interpret because the relatively long half-life of 211Pb tends to smear or average out fluctuations in q(t). If equation (7) is discretized into an n x n lnvertlble system, then one can expect to find relatively small eigenvalues, and as a result the inversion will be unduly sensitive to perturbations m the data (Wolfenbarger and Seinfeld, 1990). The inversion problem defined by equation (7) is ill-posed: solutions are not unique and solutions are unstable even if we have enough data to guarantee that a unique solution exists. Additionally, because q(t) must be nonnegative, solutions may not exist. This illposed integral equation can be successfully Inverted using regularlzatlon (Klmeldorf and Wahba, 1971; Groetsch, 1984; Wahba, 1985; Wolfenbarger and Seinfeld, 1990, 1991). The inversion code is posed as finding q(t) to minimize subject to constraints an objective function that consists of two contributions, the first being the degree of match between the observed and the predicted p data and the second being a measure of the smoothness of the signal evolution,

,=~[f~k,(t) q(t)dt-y,12+2f;(d2q(t)'~ 2 dt. \ ~ ]

(16)

The optimum q(t) we seek must both agree with the data and have some degree of smoothness or stability. The importance given to smoothness, or regularity, in the Inverted data is controlled by the value of the regularlzatlon parameter 2 (not to be confused with the decay rate constants ,~q, 22), and the solution depends on the value of 2 used. The regularization parameter 2 washes out the small ezgenvalues in the operator defined by equation (7) resulting in an approximate relationship between q(t) and epiphanlometer signal that has a well-behaved inverse. It is essential to choose an appropriate value of 2 when determining the regularized solution. If the chosen 2 is too large, then the data are undervalued and q (t) is oversmoothed; if), is too small, excessive weight is placed on the data themselves and the solution tends to exhibit nonexistent structure resulting from the errors in the data. Generalized Cross Validation (GCV) is one of the available algorithms for choosing 2 in this inversion problem (Wahba, 1975; Craven and Wahba, 1979; Golub et al., 1979, L1, 1986). Operationally, 2 is chosen to minimize V(2), P

V().)= ~

(Yt,l.a-y,) 2 w, (2),

(17)

l=l

where w,(2) are appropriately chosen weights, and YE¢, represents the ith datum pre&cted by the regularized solution when the ith measurement is omitted. The idea behind equation (17) is to choose the 2 that best enables one to predict omitted data when the data are removed one at a time from the problem. One can show (Craven and Wahba, 1979) that w,(2)=a,, II(I-Aay)II 2 V(2) = (tr [I - Aal )2,

(18) (19)

where A, is the p x p matrix that relates the measured and predicted data, tr denotes the trace operator and a, is the tth diagonal element of A ~- 1. If constraints must be included in the formulation in equations (7) and (17) to ensure the solution is nonnegatlve, then equation (l 7) must be hnearlzed for each 2 (Villalobos and Wahba, 1987; Wolfenbarger and Selnfeld, 1991).

424

~,

N,

])ANI)IS

el

a~

The second available algorithm for the choice of the regularlzatlon parameter 1~ ~h,~ minimization of the expected recovered error (CRR). Thts algorithm ~s especmlly uscml it: cases in which the standard dewatlon of the data is known. In this algomhm one c,)lculaTc, an unbiased estimate of the expected errors m the pre&cted data, ER(/)(E ts lhc cxpc, !,m(,r. operator), and the regularlzatlon parameter that minimizes ER(,). ~ can t,, ~)uz~6 (Wolfenbarger, 1990) In this paper we follow the techniques descrtbed by Wolfenbarger and Selnfeld 11990, 19911 and the code M I C R O N ts adapted to the specific eplphanlometer data mversmn problem ALGORITHM TESTING Hypothetical scenarios

In the first hypothetical test we assumed that the epiphaniometer is attempting to follow the bimodal variation in the aerosol Fuchs surface shown in Fig. 5a. The test mput has maxima at t = 6 0 and t = 180min. The instrument response was corrupted by a normal random error of 1% (Fig. 5a). The instrument response reproduces some of the features of the input signal like the two peaks with a roughly 30 mln delay. This epiphanlometer response was inverted with the above described regularlzation technique (CRR method) and the inverted data agree almost perfectly with the input signal. The effect of the epiphaniometer's error on the success of the inversion was investigated by repeatmg the above test and corrupting the instrument response by a normal random error of 5",/0 (Fig. 5b). The agreement of the inverted data with the input signal is slightly worse than m the prevtous case but remains very satisfactory. The power of the inversion technique presented here is evident m the second much more difficult test problem. The input signal has the same qualitative features (Fig. 6) but the entire variation lasts only 30 min. The epiphaniometer response corrupted again with 1% error (Fig. 6) is very weak, not capturing any of the essential features of the input signal. However, the inversion algorithm is able to reproduce almost perfectly the complicated structure of the Input signal from only this weak epiphaniometer response Application to smog chamber data

The epiphamometer was used in outdoor smog chamber experiments together with a Scanning Electrical Mobility Spectrometer (Wang and Flagan, 1990)and an OPC. An initial

l

ph.Response .. ---- Epl Inverted Signal (1~.error) 11~~ TestSlgna~~ ,

20-

C16~12

o \%

CL

\,, \

0

--

~

'

60

)

\\\

,

I

120 180 Time ( m i n )

,

"

240

,

,.X(~O

Fig 5a Input test signal to the eplphamometer, eplphanlometer response corrupted by 1% error and reverted Instrument response. The time scale of the variation is a few hour~

Aerosol data from the eplphamometer J ~J

.

~,12 '$ |

"a 8 ~

Eplph. Response (57, e r r o r ) Inverted Signal

-. . . .

t

425

!

/

i ." I b - i

/ ,,"t,-,. 1l,""/ / .' \

I,,.%

~

~

,

~

-,

,-

",t),,,' L...)" L"0

60

120 180 Time ( m l n )

240

300

Fig 5b Input test signal to the eplphanxometer, eplphanlometer response corrupted by 5% error and reverted instrument response. The time scale of the variation is a few hours

20-

~

~

--

A - -.-.-. .

Eplph. Response Inverted Signal Test Signal

/~ if" \

16-

12-

"oE 8-

0

5

10

15 Time (rnln)

20

25

30

Fig. 6 Input test signal to the eplphanlometer, eplphamometer response corrupted by 1% error and reverted instrument response The time scale of the full variation is only 15 rain

mixture of hydrocarbon (isoprene, fl-pinene or 1-octene), NO and N O 2 w a s injected into a covered Teflon bag with about 60 m 3 of clean air. After exposing the bag.to sunhght, the hydrocarbon was photochemically oxidized yielding condensible products. When a sufficiently high supersaturation was reached in the gas-phase a burst of nucleation occurred resulting in the formation of aerosol particles, which continued to grow by condensation of the available condensable material. The experiments are described in detail by Pandis et al. (1990, 1991). In the representative experiment that we shall examine here aerosol nucleation occurred about 5 min after the uncovering of the smog chamber. The Fuchs surface of the aerosol was calculated from the SEMS and OPC data using the definition of equation (11) together with expression (15). The epiphaniometer response was inverted with the algorithm described above and the two curves are presented in Fig. 7. The agreement of the two curves is satisfactory in view of the experimental errors that are associated with all the measurements revolved.

S N P~NDIS er al

426

25000 7 : = = = = DMA+OPC I # -- : -" : [ p l p h o n l o m e t e r

~E20000

~

/j

;,oooo_ o ~-J~ 0

.

.

.

.

60

.

.

.

120

180

240

Time ( m l n )

Fig. 7. Companson of the mverted epiphamometer response wRh the Fuchs surface calculated by the combined measurements of a DMA/CNC and an OPC for a representative smog chamber experiment (20 ppm of 1-octene reacting w]th 2 ppm of NOx). The Fuchs surface for the epiphaniometer is calculated by SF = 12.8 s. where s is the epiphaniometer signal.

The agreement between the inverted epiphamometer data and the Fuchs surface calculated from the DMA/OPC measurements was good for all the experiments. Summarizing these data we present in Fig. 8 a comparison of the maximum values of SF reported by the DMA/OPC with the corresponding inverted maximum values of the epiphaniometer. The best linear fit has a slope of 1.05 These results give us confidence that the epiphaniometer data with application of the proposed inversion scheme are in good agreement with the results reported by other more conventional aerosol instruments. Dependence on the choice of the regularization parameter In one of the smog chamber experiments nucleation took place at around 105 min and the raw data of the epiphaniometer suggested continuously increasing signal reaching 12,000 counts mln-1 at 230 min (Fig. 9). These raw data were inverted by using both CRR and 25000 leloo

MO

E .2,°2oooo O

15000 u

10000-

"'~

E o "E 5000

,"

0

s

0

0

' 5000 10()00 15000 ' 2 0 ( ~ ' 25000 DMA+OPC Fuchs' Surfoce ( c m "~)

Fig. 8 Comparison of the maximum values ofS F reported by the DMA/OPC with the correspondmg inverted maximum values of SF from the epiphaniometer. Also shown, the linear regression and the line for SF (DMA/OPC)= SF (eplphamometer)

Aerosol data from the eplphanlometer

427

100000 Method. Std Dev

-- err; 80000

-

I

20+ ~ + O 0 0 1 n

I

egcv; 2 0 + ~rfi+O O01n •

-.- Raw Data

~

/

'

1

I

,

E 80000

tt~ r~

40000

C 0 L)

20000

b ..........

iio .........

12o

Time

lao

240

(rain)

Fig 9 Comparison of the C R R with the C G C V algorithm for the inversion of smog chamber data The raw epiphamometer data are also shown

CGCV for the choice of the regularization parameter 2 (Wolfenbarger, 1990). The data uncertainty used for the CRR method was the sum of three components: a constant one, one varying like the square root of the number of measured counts per interval, n, and a linear one varying like 0.1% of n. Under these conditions CRR had no problem inverting the data and the inverted curve was in good agreement with the D M A / O P C data. The inversion of the data indicated correctly that the aerosol Fuchs surface reached a maximum at around 175 rain and started declining due to particle losses to the walls. The CGCV algorithm on the other hand chose a small regularization parameter resulting in a highly oscillatory inverted signal. These results suggest that the CRR algorithm performs much better than the CGCV algorithm for the inversion of the epiphaniometer data. The dependence of the CRR response on the choice of the data uncertainty is demonstrated in Fig. 10. In this smog chamber experiment a significant burst of nucleation occurred at around 20 min resulting in a rapid increase of the aerosol Fuchs surface. Use of the CRR method with only the constant and the square root component in the data uncertainty expression results in significant oscillations after the rapid increase of the Fuchs surface area. The oscillations decrease in size with the addition of a 1% linear error lO000O

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component, decrease even turther for 2°0 and disappear completel 3 for a :", . ' ~ , ; Therefore. for a very rap~d increase m Fuchs surface lmkethe present one resulting m a drastic change of the second derivative, one may be obhged 1o increase the data unce~a',n~5 allowing CRR 1o choose a larger value of the regularizatlon paramete~ / resultli~a ~ , smoother cur~ c CON(

LUSIONS

The eplphanlometer has been represented as a plug flow reactor an which lead clusters attach to th. entering aerosol particles, decay radioactively and thei~decay ~s measured b~ya radiation de~cctor. The response was tested against the experimentally obtained instrument response to a step function and was found satisfactory. The Fuchs surface of an aerosol particle was defined and a simplified formt~Ia to~ its calculation was proposed based on epiphanlometer calibration data The instrument response function was then used in an inversion algorithm originally developed tor the inversion of aerosol size distribution data obtained by more conventional aerosol instruments. The algorithm ms based on the regulanzation technique in which the optmaum solution both agrees with the data and has some degree of smoothness or stabdlt? The algorithm was tested successfully m hypothetical scenarios and was able e~,en to resolve fluctuations of the aerosol Fuchs surface taking place over the order of a few minutes The inverted eplphaniometer data, obtained during outdoor smog chamber experiments° were compared to the Fuchs surface calculated from D M A / O P C data from the same experiments and the agreement was again satisfactory The range of the eplphanlometer applications has therefore been extended to phenomena that take place over time scales of a few minutes Also, a method is proposed and tested that enables one to compare directly the eplphaniometer results with those of other more traditional aerosol instruments Acknowledyemenls---We thank S E Paulson for her help m perfornung the smog chamber experiments q h~.~~ork was supported by the Swiss National Science Foundation

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