termine dissolved sulfite oxidation, equilibrium SO2 partial pressure, and magnesium sulfite saturation.

Figure 5 indicates that, for the measured pH of 5.3, a sul- fur-to-magnesium ratio of 0.80/0.70 or 1.14, and a dissolved magnesium concentration greater than 0.4 molal, the value of (1/1 + Ox) is about 0.62. This corresponds to 61% oxida- tion.

Figure 3 has a reference pH of 5.0. For this reference pH and 61% oxidation, linear interpolation of magnesium and SO2 pressure values gives an SO2 partial pressure of 3.3 X 10-4 atm for the measured magnesium molality of 0.70. The pH cor- rection factor given in Table I11 for a pH of 5.5 is 0.2; therefore, the correction factor for a pH of 5.3 is 0.2 to the power of (5.3 - 5.0)/(5.5 - 5.0) = (0.2)0.6 = 0.38. The equilibrium SO2 par- tial pressure for the liquor sample is (3.3 X 10-4)(0.38) = 1.25 x atm.

To determine sulfite saturation from Figure 4, the SO2 partial pressure must be corrected to a pH of 5.0 as indicated in Table 111. The correction factor for a pH of 5.5 is 0.15; therefore, the factor for the measured pH of 5.3 is (0.15)0.6 = 0.32. The SO2 partial pressure to be used in Figure 4 is (1.25 X 10-4)/0.32 = 3.9 X atm. By linear interpolation the sulfite saturation is found to be 80% for the sulfite oxidation of 61%.

By comparison with these graphical values of 61% oxidation, 1.25 X atm SO2 partial pressure, and 80% magnesium sulfite saturation, the chemical model gives calculated values of 63% oxidation, 1.24 X atm S02, and 78% saturation.

Acknowledgment This paper was prepared while working on the EPA-funded

Shawnee wet scrubbing Test Program, with the support and encouragement of John E. Williams, Project Officer. We ap- preciate the helpful comments of Harlan N. Head, Nancy E. Bell, and Kenneth A. Strom of Bechtel National, 1nc.k Air Quality Group, and also those of Gary T. Rochelle of the University of Texas at Austin. We are grateful for the assis- tance of Eleanor L. Tape1 and Jackson Hing. Literature Cited (1) Quig, R. H., “Chemic0 Experience for SO1 Emission Control on

Coal-Fired Boilers”, Coal and the Environment Tech. Conf., Louisville, Ky., Oct. 1974.

(2) Quigley, C. P., Burns, J. A., “Assessment of Prototype Operation and Future Expansion Study-Magnesia Scrubbing a t Mystic

Generating Station, Boston, Massachusetts”, EPA Symp. on Flue Gas Desulfurization, Atlanta, Ga., Nov. 1974.

(3) Clement, J. L., Tappi, 49 (8), 127A-34A (1966). (4) Cronkright, W. A., Leddy, W. J., Enuiron. Sci. Technol., 10 (6),

(5) Head, H. N., Bechtel Corp., “EPA Alkali Scrubbing Test Facility: Advanced Program-Third Progress Report”, EPA Rep. 600/7-

(6) Rochelle, G. T., PhD thesis, University of California, Berkeley, Calif., 1976.

(7) Smith, W. T., Parkhurst, R. B., J . Am. Chem. SOC., 44,1918-27 (1922).

(8) Hagisawa, H., Bull. Inst. Phys. Chem. Res. (Tokyo) , 12,976-83 (1933).

(9) Conrad, F. H., Brice, D. B., Tappi, 70,2179-82 (1948). (10) Kuzminykh, I. N., Babushkina, M. D., J. Appl . Chem. USSR,

(11) Semishin, V. I., Abramov, I. I., Vorotnitskaya, L. T., Khim.

(12) Pinaev, V. A., J . Appl . Chem. USSR, 36 ( lo) , 2049-53 (1963). (13) Pinaev, V. A., ibid., 37 (61, 1353-5 (1964). (14) Markant, H. P., McIlroy, R. A., Matty, R. E., Tappi, 45 (11),

849-54 (1962). (15) Pyle, R. E., “Experimental Results for the Equilibrium Stidies

on MgSO, Hydrates”, Radian Corp. Tech. Note 200-045-36-04, 1976.

(16) Robinson, R. A., Stokes, R. H., “Electrolytic Solutions”, 2nd ed., London, England, 1965.

(17) Markant, H. P., Phillips, N. D., Shah, I. S., TAPPI= 4- (ll),

(18) Rabe, A. E., Harris, J. F., J . Chern. Eng. Data, 8, 333-6

(19) Bromley, L. A., AICHE J. , 19 (2), 313-20 (1973). (20) Tartar, H. V., Garretson, H. H., J . Am. Chem. Soc., 63,808-16

(21) Johnstone, H. F., Leppla, D. W., ibid., 56, 2233-8 (1934). (22) Lowell, P. S., Ottmers, D. M., Strange, T. I., Schwitzgebel, K.,

De Berry, D. W., “A Theoretical Description of the Limestone In- jection-Wet Scrubbing Process”, Final Rep. for EPA Contract No. CPA-22-69-138 with Radian Corp., 1970.

(23) Nair, V.S.K., Nancollas, G. H., J . Chem. Soc. (London), 1958, pp 3706-10.

(24) Stock, D. I., Davies, C. W., Trans. Faraday SOC., 44, 856-9 (1948).

(25) Harned, H. S., Owen, R. B., “The Physical Chemistry of Elec- trolytic Solutions”, 3rd ed., Reinhold, New York, N.Y., 1958.

(26) Lewis, G. N., Randall, M., Pitzer, K. S., Brewer, L., “Thermo- dynamics”, 2nd ed., McGraw-Hill, New York, N.Y., 1961.

(27) Pitzer, K. S., J . Chem. Soc. Faraday Trans. I I , 68, 101-13 (1972).

(28) Kester, D. R., Pytkowicz, R. M., Lirnnol. Oceanogr., 14,686-91 (1969).

569-72 (1976).

77-105, 1977.

30,495-8 (1957).

Khim. Tekhnol., 2,834-9 (1959).

648-53 (1965).

(1963).

(1941).

Receiued for review February 10, 1978. Accepted April 28, 1978.

A General Method for Fitting Size Distributions to Multicomponent Aerosol Data Using Weighted Least-Squares

Otto 0. Raabe Radiobiology Laboratory and Department of Radiological Sciences, School of Veterinary Medicine, University of California, Davis, Calif. 95616

This information is directed to investigators using multi- stage and other multicomponent aerosol samplers for studies of aerosol size distribution. A general and straightforward approach is described for fitting a chosen size distribution (probability densityj function to the grouped data collected with such samplers as cascade impactors, cascade centripeters, horizontal elutriators, aerosol centrifuges, multiple cyclone samplers, and other devices that fractionate aerosol samples into several components or collect samples representing sev- eral different size ranges. The actual collection efficiencies with respect to particle size of each component or stage are

used to fit the size distribution function by weighted nonlinear least-squares regression analysis with each datum weighted by the reciprocal of its estimated variance. Particle size may be geometric, aerodynamic, or whatever size convention is measured by the sampling device. Estimates are made of the confidence limits of the fitted parameters, and the correlation coefficient and chi-square values are used to test the accept- ability of the resulting size distributions. Illustrative examples are given for fitting log-normal functions to data collected with a cascade impactor, with a spiral-duct aerosol centrifuge, and with a diffusion battery.

1162 Environmental Science & Technology 0013-936X/78/0912-1 162$01.00/0 @ 1978 American Chemical Society

Various methods and instruments are available for col- lecting size-fractionated or otherwise size-discriminated samples of an aerosol for evaluation of its size distribution. The multicomponent aerosol data may be overlapping if parallel samples are taken, or censored if some portion of the distribution is not sampled. By having several unique com- ponents of aerosol data, it is possible to evaluate the form of the size distribution from which the data were collected.

Such measurements of aerosol size distribution are essential for understanding and evaluating the details of the behavior of an aerosol and in documenting observed characteristics. Raabe ( I ) describes the nature of such data and their con- version to normalized form to provide estimates of the prob- ability density function, which describes the particle size distribution. Whatever the methods or instruments used, it is useful to fi t a suitable mathematical function having rela- tively few characteristic parameters to yield a satisfactory mathematical description of the probability density function. In addition, it is essential to estimate the statistical confidence limits for the fitted parameters, to test the hypothesis that the fitted function and the true size distribution (as represented by the data) are the same, and to provide other information concerning the statistical reliability of the fitted function. These statistical tests separate the “eyeball” fits from those obtained by the rigorous methods used in this report and provide meaningful comparisons between observed size dis- tributions.

In the analysis of microscopically determined size data, Kottler ( 2 ) used a fitting procedure closely related to weighted least-squares and called the method of chi-square minimum. Kottler did not incorporate collection efficiencies into his analysis. Preining ( 3 ) described the essential features of the effect of overlap of stage efficiencies on fitted size distribu- tions, and Geisel(4) developed an approximate simultaneous equation method using simple sampler efficiency functions to estimate size distributions. Statistical evaluations were not provided.

This report describes a general method for fitting and testing the fit of size distribution (probability density) func- tions to multicomponent aerosol data using the well-known and established methods of nonlinear regression analysis, commonly called weighted least-squares. A preliminary ver- sion of this approach was introduced by Raabe and Tillery ( 5 ) for fitting cascade impactor data to log-normal size distribu- tions. This method can be used with data from all types of instruments with known collection efficiencies and with any desired size distribution function (including bimodal and trimodal functions) if the number of data components exceeds the number of unknown parameters. Because the actual ef- ficiency function associated with each sampler stage or data component is used directly, this technique is called the stage efficiency least-squares method for fitting size distribution functions to data.

Methods Size Distribution Functions. Various choices may be

made of a suitable size distribution function (probability density), f ( D ) , to describe aerosol size distributions as a function of particle size, D . This normalized function, with units of fraction of the distribution per unit size interval ( I ) , is applied to the population of particles by integrating over some specific range of sizes, D 1 to D s , to yield the fraction of the population in the range. When f ( D ) is integrated with respect to size over all sizes, the result is unity.

As an example, one function that has proved particularly useful in particle size analysis as a probability density is the log-normal function (1). The log-normal size distribution, defined as a distribution of sizes whose logarithms are nor- mally distributed (Gaussian), is described linearly by:

with a1 the median diameter of the distribution (geometric mean) and a2 the geometric standard deviation. In this case, a1 and a2 are the parameters that need to be determined to fit this function to aerosol data.

For some aerosol distributions a function created by the sum of two or more log-normal functions to create bimodal or multimodal forms might describe the data better. In general, a total of m characteristic parameters, a,, will describe the chosen function. These parameters will have to be determined in the fitting procedure, so at least m + 1 unique datum values must be available to provide one degree of freedom.

Multicomponent Aerosol Data. Grouped or size-frac- tionated aerosol data can be obtained with various instru- ments, such as inertial samplers, settling or diffusion batteries, aerosol spectrometers, electrical mobility analyzers, light scattering photometers, light microscopes, and electron mi- croscopes. The inertial samplers commonly used are impactors (6, 7), centripeters (8) , virtual impactors (9 ) , and cyclones (10-12); these four provide size-separated samples based on aerodynamic size characteristics (13). The stages of such de- vices in series are usually referred to as in cascade (6, 7), whereas some arrangements place the collectors in parallel so that each operates independently ( I O ) . Settling batteries, such as elutriators (14), and aerosol spectrometers, such as the conifuge (15) or the spiral-duct aerosol centrifuge invented by Stober and Flachsbart (16), also provide size-separated samples based on aerodynamic size. Diffusion batteries ( I 7) provide multicomponent aerosol data based on particle dif- fusion coefficients, which are related to the geometric size characteristics of the aerosol particles. Electrical mobility analyzers (18) size particles on the basis of electrical mobility characteristics. Light scattering particle counters (19) respond to both particle size and refractive index. Microscopy with either light or electron microscopes yields grouped size data based on geometric considerations, such as projected area diameter (1 ) .

Multicomponent data can be represented as the fraction of the aerosol in each of n size classes, groups, or sample stages. The fractional relationship will depend on the exact nature of the sampling method as to whether mass, radioactivity, chemical concentration, particle surface, or simply particle number is determined for each data component. Further, the conventions by which particle size is defined (or related to the instrumental response) and the exact relationship of the data components vary depending on the physical principles used to collect the samples and the orientation and interrelation- ship of the collectors. In any case each of n components of data can be expressed as an observed fraction, F,*, of the total distribution based on quantity of the aerosol collected in sample component i.

The fraction of the aerosol in a given component will depend on the relative collection efficiency, E, ( D ) , with respect to particle size for the n instrument stages or chosen size range components. The particle size, D , may be aerodynamic di- ameter in the case of inertial samplers, geometric diameter in the case of diffusion batteries, mobility diameter in the case of mobility analyzers, or other appropriate size convention. This efficiency function is dimensionles with values between zero and unity describing the probability of a particle of size, D , being separated in component, i. When E, (D) is known for each of the n data components (the efficiencies need not be explicit functions), the fraction, F,, to be found in a given stage, i , for a chosen size distribution function, f ( D ) , can be predicted. For example, if an instrument has stages in cascade, with the first stage in the series numbered 1 and the last stage numbered n (collecting particles that negotiate the other n

Volume 12, Number 10, October 1978 1163

- 1 stages), the fraction, Fi, expected in stage i is given by:

with Eo(D) = 0 for all sizes (no stage zero). The product term containing Ek ( D ) corrects for particles that are not available to stage i because they were co ected on an earlier stage.

If the stages of a sample are in parallel or if an aerosol spectrometer is used, the fraction expected in a given sample component is given more simply by:

Even though the required integration to provide a value of F, will not be explicit, standard numerical methods (such as the trapezoidal rule) can be employed using digital computers to perform the calculation.

Fitting Mathematical Functions with Weighted Least-Squares. The use of weighted least-squares for fitting functions to data is a well-established method with several nonlinear techniques available (20-23). If the normal law of errors is assumed to apply to datum values, the probability, L,, associated with an observation of a fraction, F,*, of aerosol in data component (sample stage) i when the expected value is F, should be:

(4)

with u1*2 the variance of the difference between the observed value, F,*, and the corresponding true value, F,. The total probability, L , for all the observed n datum values is therefore the product of the n individual probabilities and is:

L = e-(Ft* - F1)2/2uz*2

L = , -1/2z,l l[(F,*-F,)*/u~*zl ( 5 ) When L is a function of observed sample data with s , * ~ the data based estimates of g,**, it is called the likelihood func- tion. This likelihood function, L , is maximized when the summation in the exponential term is minimized:

As F, is a function of f ( D ) , minimal exponential sums will be achieved with proper choice of the characteristic parameters, ai, of the size distribution function. The values of the recip- rocal of si*2 are commonly called the weighting factors, and the procedure for minimizing this sum is called weighted least-squares. The maximum likelihood is obtained when the characteristic parameters of the distribution function, f ( D ) , are chosen such that the partial derivative of the expression to be minimized is equal to zero with respect to each param- eter. This computation is given by a family of derivatives:

for j = 1 through m for calculated Fl(a l , a*, . . . ,CY,,,). Values of dF, ldaJ are needed for each aJ for this calculation.

In general, the function, F, (a l , 0 2 , . . . ,CY,,, 1, as given by Equations 2 or 3 (or as otherwise defined) will not be explicit. But, numerical methods for nonlinear least-squares regression analysis do not require explicit functions if single values of the function can be computed to a desired level of accuracy. Likewise, the values of d F , / d a J required for solutions of Equation 7 need not be explicit if individual values of the derivatives can be computed.

The general theory for nonlinear regression analysis has been well described (20,23). Numerical methods are described by Bevington (21) and by Daniel and Wood (22). The nu-

merical system used in this report is essentially as described by Lietzke (24) . Briefly, a first order Taylor approximation is made of the function describing F, in terms of estimates of the m values of aJ and incremental changes in these values, AaJ. If the changes are small, the partial derivatives of the weighted sum of squares (Equation 6) with respect to the AaJ values set equal to zero are the equations providing the proper ACY] values to correct the original estimates of CY, and provide the maximum value of L (Equation 5 ) . The result is a family of equations:

with j = 1 through m parameters, which yields a family of m solvable simultaneous equations as dF,/d(Aa,) = bF,/daJ. A matrix inversion routine is used to solve these equations. This procedure for calculating improved estimates of the weighted least-squares values may be repeated as many times as required to provide the desired accuracy.

Statistical Evaluation. As part of the least-squares pro- cedures, estimates are calculated of the variance associated with each fitted parameter, CY], based on the variability asso- ciated with the data after removal of variability due to re- gression. These estimated variances provide standard error values, which indicate the importance (or lack of importance) of each cyJ in the chosen distribution function and provide a basis for comparison to other fitted distributions. Also cal- culated are estimates, s, 2, of u, *, the variance associated with each fitted value F,, which equals the variance of the differ- ence between a computed value of F, and the corresponding true value of F,.

Using s, the validity of the hypothesis that the parent population from which the sample was taken is distributed as f ( D ) is conveniently tested using a chi-square test of the deviations of the data values of F,* from the theoretical ex- pectations F, based on the fitted function. The value of chi- square for this test is given by:

The well-known theoretical distribution of such chi-square sums in random sampling is contingent on the number of degrees of freedom in the test. The n data components and m calculated parameters yield ( n - m ) degrees of freedom. Tabular values of chi-square for various numbers of degrees of freedom and various confidence levels are found in statis- tical tables and texts (20).

Calculation of the correlation coefficient, R , associated with the regression analysis and performing a t-test as described by Anderson and Bancroft (20) will test further the adequacy of the chosen distribution function by testing the hypothesis that the function and the data are not correlated ( R = 0). This is a test of goodness of fit, since R 2 measures the proportion of the total variance accounted for by the fitted function. The tabular values of t are also dependent on the number of de- grees of freedom in the test; the n data components and m calculated parameters yield (n - m ) degrees of freedom.

Special Cases. Some multicomponent data may be in- complete, censored, or overlapping. The general method can accommodate these types of data sets without alteration, if the fractions of the total aerosol associated with each sample can be estimated and the number of data components exceeds the number of characteristic parameters of the chosen aerosol size distribution function. Statistical results will automatically be reflected in the standard error associated with each of the fitted parameters, ai.

If a portion of the aerosol size range cannot be measured and the effect of that portion on the fractional values F,* cannot

1164 Environmental Science & Technology

be readily assessed, the data values are the actual amounts collected on each stage or in each size group. In this case the chosen size distribution must be multiplied by a factor equal to the unknown total amount of aerosol in the whole sample distribution. This factor becomes an additional a, parameter, but the procedure is otherwise unchanged.

When the data are censored or truncated, the function de- scribing the expected amount, A,, on each stage (in each data component) is compared to the data values A,* with:

A, = @,+lFl (10)

and f i t instead of F, to yield a fitted value for as well as for the m parameters in the size distribution function. The parameter, a,+1, then becomes an estimate of the unknown total amount of the aerosol that would have been in the sample if it were not truncated.

Examples Cascade Impactor. The stage-efficiency least-squares

method has been tested most extensively with cascade im- pactor data. In application of the general method to such data, the stage efficiency function for each impactor stage must be known for use in Equation 2. As cascade impactor operation has been studied extensively, both theoretical (25) and ex- perimental efficiency curves are generally available for most instruments. Although these stage-efficiencies are not explicit mathematical functions, they can be satisfactorily estimated with tabular values and numerical interpolation. Particles on cascade impactor samples are analyzed by mass or some other measure of quantity collected with respect to aerodynamic diameter (13), and the size distribution (probability density) functions will have units of fraction of mass (or related quantity) per unit aerodynamic diameter size interval.

A total of 50 samples analyzed with this method were compared with ,each other and to graphical analysis using logarithmic probability graph paper. These data were col- lected for five different seven-stage round-jet cascade im- pactors having self-contained back-up filters (7). The effi- ciency curves of the stages were each divided into 23 linear segments for the required numerical integration. Log-normal mass distribution functions were fit to the data. In one ex- ample (Figure 1) a sample with a fitted mass median aerody- namic diameter ( M M A D ) of 1.59 f 0.08 (SE) pm and geo- metric standard deviation (a,) of 1.74 & 0.08 (SE) showed a chi-square value for this fit of 3.7, indicating a good fit to the

A 06- 1

9 CASCADE IMPACTOR DATA

, , , , , , 0 0 0 5 1.0 1 5 2 0 2 5 3 0 3.5 4 0 4.5 5.0

AERODYNAMIC D IAMETER ( pm I

Figure 1. Example of log-normal aerosol mass distribution with respect to aerodynamic diameter (pm) as fit to data collected with a seven-stage cascade impactor [mass median aerodynamic diameter (MMAD) = 1.59 f 0.08 (SE) p m and geometric standard deviation (ag) = 1.74 f 0.08 (SE)] Efficiency functions for each stage, El through E,, and back-up filter (E8) and corresponding distributions of fractions F1 to F8 are shown. Fractions are given by integral area under each curve

available data being within one standard deviation of the expected value. The largest chi-square value acceptable a t the 1% significance level is 16.8.

For the fifty data sets, the ratio of the MMAD determined graphically to the one determined with the least-squares stage-efficiency method had a mean of 0.979 (f0.043 SD; f0.006 SE). The corresponding ratio of geometric standard deviations (ag) had a mean of 1.062 (f0.085 SD; f0.012 SE). The graphical method underestimates the MMAD by a t least 2.1% and overestimates the ar by a t least 6.2% for half of the samples. The range of variability of individual estimates by the graphical method is so large that the MMAD is overesti- mated in 31%, and ag is underestimated in 23% of such sam- ples. The graphical method yields widely variable results, which are consequently somewhat unreliable.

In contrast, the coefficient of variation (the ratio of the standard error to the mean) of the least-squares-fitted MMAD and ag values averaged 6.2 and 5.2%, respectively. Of the fifty sets, all but eight had chi-square values less than 16. When the seven sets with the largest chi-square values were eliminated, the average chi-square value of the remainder was 6.6, and the observed variance was 13.2 as compared with the theoretically expected values of 6 and 12, respectively, for samples collected from parent populations that are log-normal.

Diffusion Batteries. The stage-efficiency least-squares method is particularly useful for analysis of data collected with diffusion batteries because the efficiency of collection (or transmission) of particles through a diffusion battery can be described by an easily evaluated analytical function (17). The cutoff characteristics of these instruments are not sharp, and other methods of data analysis tend to be inaccurate or of limited usefulness. With a diffusion battery, the penetration of aerosol through individual stages is measured either as mass fraction or as number fraction with respect to geometric di- ameter (which determines the particle diffusion coefficient) of spherical particles. Hence, the probability density may be either mass or number distributions with units of fractions of distribution per unit geometric size interval.

Data collected with a portable diffusion battery as described by Sinclair (26) have been analyzed. This diffusion battery consisted of collimated perforated structures of various lengths and each containing 14500 holes of 0.009 in. diameter. The fractional penetrations of an ambient outdoor aerosol between stages in series were determined with a condensation nuclei counter. As the efficiency of penetration of a given size depends on the penetration through all stages in series before the point of measurement, the efficiency functions in this example (Figure 2) are the products of efficiency function of the individual stages. For this natural aerosol the fitted count mediam diameter ( C M D ) was 0.0239 f 0.0012 (SE) pm, and ag was 2.34 f 0.16 (SE). The chi-square was only 3.3, and the correlation coefficient was 0.998. A prototype computer pro- gram for this method was provided to and used by Sinclair (26).

Spiral Duct Aerosol Centrifuge. An aerosol sample is collected in a spiral duct aerosol centrifuge (16) on a long narrow foil strip with particles separated according to their aerodynamic diameters. Such a sample is cut into several pieces, and the mass of particles (or related quantity of aero- sol) is determined for the specific size range represented by each segment. As with impactors, the size distribution will have units of fraction of mass per unit aerodynamic diameter size interval. However, a portion of the size distribution may not be collected in the centrifuge, and the data may be cen- sored, in which.case Equation 10 must be used.

Samples were collected with a spiral-duct aerosol centrifuge, the Lovelace Aerosol Particle Separator (LAPS), using methods described by Raabe et al. (27). In this unit one sample is collected on a back-up filter, so that the quantity of

Volume 12, Number 10, October 1978 1165

1 T i - in this report is applicable to the various types of multicom- ponent aerosol data as commonly collected with cascade im- pactors, diffusion batteries, and other instruments that fractionate an aerosol sample with respect to particle size. Based on the well-known and efficient procedures of nonlinear least-squares regression analysis, the method provides esti- mates of the statistical reliability of the results, which are useful in comparing the results of different size analyses. Because this method accommodates the actual efficiencies in the separation of each size of particle, approximations of ef- ficiency, such as use of ideal (but incorrect) cut sizes or ef- fective cutoff diameters, are avoided. The overall accuracy of the results will, of course, depend in part on the accuracy of the experimentally or theoretically determined collection efficiency functions, but the stage-efficiencies least-squares method makes beneficial use of the best calibration values available.

The use of the least-squares method for fitting functions can be contrasted to the minimum chi-square method of Kottler (2). In the least-squares method the available infor- mation concerning the reliability of the data is used to esti- mate variances (Equation 6) to yield the maximum likelihood estimates (Equation 5). On the other hand, chi-square (Equation 9) is not expected to be minimum in random sam- ples, and is reserved herein for tests of the statistical adequacy of fitted functions.

The examples given in this report employ log-normal par- ticle size distributions. The general usefulness of the log- normal function is well-established. However, other distri- bution functions such as described by Silverman et al. (28) and Herndan (29) can also be used with this technique. Also, the method can be extended to use multicomponent functions fitted to the data from one instrument or to data collected over an extended range of sizes by more than one instrument.

It will be noted in the illustrations (Figures 1-3) that there is no distinct upper limit in particle size data. The cascade impactor (Figure 1) accumulates all larger particles together on the first stage. The diffusion battery (Figure 2) passes the larger particles to the last sample component, the spiral duct centrifuge collects all the larger particles together a t the be- ginning of the channel, and may have entrance losses as well. This is not a shortcoming of the method since the stage-effi- ciency least-squares procedure properly treats the mathe- matical implications of these imprecisions with respect to the larger particles. If a more detailed analysis is required of the distribution of the larger particles, different operating con- ditions need to be chosen for the aerosol sampling devices.

The power and usefulness of this method are best illustrated by the case of the diffusion battery. With this method, the data collected with diffusion batteries were used accurately in the fitting process even though the efficiency functions were not sharp. In fact, because the efficiency curves for this in- strument are well defined and relatively insensitive to small changes in sampling conditions, diffusion batteries are among the more dependable instruments for sizing submicrometer aerosols.

Computer programs have been developed to apply the stage-efficiency least-squares method to data collected with various sampling instruments. These programs in Fortran IV are available from the author upon request.

Acknolcledgment The author is indebted to Leon Rosenblatt and E. Bradley

for review of the manuscript and for useful suggestions, to Ken Shiomoto and Shirley Coffelt for preparation of the illustra- tions, and to J. Azevedo for editing the manuscript.

Nomenclature CY, = parameters with j equal 1 through m in mathematical

c_ - 2 5 1 ,/ \ DZFFU§ZON EATTERI DATA z:

; 0 5 'Obss* E e

E. 0 0 001 0 0 2 003 004 005 006 007

GEOMETRIC D IAMETER Ipml

Flgure 2. Example of log-normal aerosol number distribution with re- spect to geometric diameter for spherical particles as fit to data col- lected with eight-stage diffusion battery [count median diameter (CMD) = 0.0239 f 0.0012 (SE) km, and geometric standard deviation ( u g ) = 2.34 f 0.16 (SE)] Efficiency functions of penetration through each stage, E , through E8, and corresponding distributions of penetration fractions, F, to F8. are shown. Fractions are given by integral area under each curve

- 0 8 . E SPIRAL DUCT CEh'TRIFUG€ DATA

3 0 7 n 0 6 1

L

-,--DATA HISTOGRAM I /

0 I 2 3 4 5 6

AERODYNAMIC D I A M E T E R ipml Figure 3. Example of log-normal aerosol mass distribution with respect to aerodynamic diameter as fit to data collected on each of 22 segments of collection foil plus back-up filter of LAPS spiral duct aerosol cen- trifuge [mass median aerodynamic diameter (MMAD) = 2.09 f 0.05 (SE) I.tm and geometric standard deviation (ug) = 1.55 f 0.03 (SE)] ideal (100%) efficiency is assumed for each value F,' on given segment, and corresponding calculated value F, is based on fitted aerosol mass distribu- tlon

particles smaller than those collected on the foil is measured also. Some large particles may not have been collected in the centrifuge but were assumed to be nil for this example. The collection foil was cut into 22 segments for mass analysis (27). For each segment the efficiency function is assumed to be ideal, equal to 100% over the size range of the segment and zero for all other sizes. Entrance losses could have been assigned with a smaller efficiency for each segment. The fractions of the aerosol sample, F,*, on each segment were used to fit the calculated fractions, FL, as given by Equation 9. A log-normal function was used to describe the mass distribution. The re- sulting MMAD and a, in one example (Figure 3) were 2.09 f 0.05 (SE) K r n and 1.55 f 0.03 (SE), respectively. The chi- square value was only 9.9 for this sample having 21 degrees of freedom, and the correlation coefficient was 0.98. The probability that this correlation is not significant is less than 2%.

Discussion The general stage-efficiency least-squares method described

1166 Environmental Science & Technology

description of distribution function (probability density), f ( D )

U, 2 = the variance of the difference between a value of F, for the computed regression function and the corresponding true value of F,

= the variance of the difference between the observed datum value, Fl*, and the corresponding true value

ug = the geometric standard deviation, a characteristic pa- rameter of a log-normal particle size distribution function; fitted values are estimates with standard errors provided

x2 = the chi-square distribution used for analysis of variance and testing of functions

A, = the calculated quantity of aerosol in component (or sampler stage) number i; based on the collection efficiency E, ( D )

A,* = observed (collected) quantity of aerosol in data com- ponent (or sampler stage) number i

CMD = count median diameter, the characteristic param- eter in a log-normal particle size distribution function de- scribing the particle number distribution with respect to geometric diameter; fitted values are estimates with stan- dard errors provided

D = characteristic dimension used to describe particle size, such as particle diameter

E, ( D ) = dimensionless collection efficiency function (with values from zeto to unity) for component (sampler stage) number i with respect to particle size D

f ( D ) = a particle size distribution (probability density) as function of size D with units fraction of distribution per unit size interval

F, = fraction of aerosol size distribution expected to be in component (or sampler stage) number i based on the chosen (fitted) distribution function

F,* = observed (collected) fraction (data component) of aerosol size distribution on sample stage number i with i = 1 through n’such components

1 = subscript index of data component (sampler stages) j = subscript index of parameters in distribution function,

f ( D ) L , = a probability associated with a datum observation L = a probability associated with a group of data observa-

tions n = number of data components or sampler stages m = number of parameters in distribution function, f ( D ) M M A D = the mass median aerodynamic diameter, a char-

acteristic parameter of a log-normal size distribution function describing particle mass distribution with respect to aerodynamic diameter. Fitted values are estimates with standard errors provided

R = correlation coefficient relating the fitted distribution function to the aerosol data

s,2 = estimate of the variance, u, 2, for the fraction F , based

on the fitted distribution function f ( D ) and determined in the nonlinear weighted least-squares regression analysis

si* * = estimate of the variance of the difference between the observed datum value of Fi* and the true value as deter- mined from statistical analysis methods as associated with measurements of Fi*

Literature Cited (1) Raabe, 0. G., Aerosol Sci., 2, 289 (1971). (2) Kottler, F., J . Franklin Inst., 251,499 (1951). (3) Preining, O., J . Rech. Atmos., 2-3, 145 (1966). (4) Geisel, W., Staub-Reinhalt. Luft, 28, 25 (1968). (5) Raabe, 0. G., Tillery, M. I., Am. Ind. Hyg. Assoc. J . , 29, 102

(6) May, K. R., J . Sci. Instrum., 22, 187 (1945). (7) Mercer, T. T., Tillery, M. I., Newton, G. J., Aerosol Sci., 1, 9

(1970). (8) Hounam, R. F., Sherwood, R. J., Am. Ind. Hyg. Assoc. J. , 26,112

(1965). (9) Loo, B. W., Jaklevic, J. M., Goulding, F. S., in “Fine Particles”,

B.Y.H. Liu, Ed., pp 311-50, Academic Press, New York, N.Y., 1976.

(10) Lippmann, M., Harns, W. B., Health Phys., 8,155 (1962). (11) Lippmann, M., Kydonieus, A., Am. Ind. Hyg. Assoc. J. , 31,730

(12) Lippmann, M., Chan, T. L., ibid., 35,189 (1974). (13) Raabe, 0. G., J . Air Poilut. Control Assoc., 26,856 (1976). (14) Walton, W. H., Brit. J . Appl. Phys., Suppl. 3, S29 (1952). (15) Sawyer, K. F., Walton, W. H., J . Sci. Instrum., 27, 270 (1950). (16) Stober, W., Flachsbart, H., Enuiron. Sei. Technol., 3, 1280

(1968).

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(1969). (17) Thomas, J. W., J . Colloid Sei., 10,246 (1955). (18) Whitby, K. T., Clark. W. E., Tellus, 17,573 (1966). (19) Hodkinson, J. R., in “Aerosol Science”, C. N. Davies, Ed., pp

(20) Anderson, R. L., Bancroft, T. A., “Statistical Theory in Re-

(21) Bevington, P. R., “Data Reduction and Error Analysis for the

(22) Daniel, C., Wood, F. S., “Fitting Equations to Data”, Wiley-

287-357, Academic Press, New York, N.Y., 1966.

search”, McGraw-Hill, New York, N.Y., 1969.

Physical Sciences”, McGraw-Hill, New York, N.Y., 1969.

Interscience. New York. N.Y.. 1971. (23) Draper, N. R., Smith,”., “Applied Regression Analysis”, Wiley,

New York, N.Y., 1966. (24) Lietzke, M. H., “A Generalized Least Squares Program for the

IBM 7090 Computer”, ORNL-3259, Oak Ridge National Labora- tory, Oak Ridge, Tenn., 1962.

(25) Marple, V . A., thesis, COO-1248-21, University of Minnesota, Minneapolis, Minn., 1970.

(26) Sinclair, D., Am. Ind. Hyg. Assoc. J. , 33, 729 (1972). (27) Raabe, 0. G., Boyd, H. A., Kanapilly, G. M., Wilkinson, C. J.,

Newton, G. J., Health Phys., 28,655 (1975). (28) Silverman, K., Billings, C. E., First, M. W., “Particle Size

Analysis in Industrial Hygiene”, Academic Press, New York, N.Y., 1971.

(29) Herndan, G., “Small Particle Statistics”, Academic Press, New York, N.Y., 1960.

Received for review August 23,1977. Accepted May 2, 1978 Research supported by the Diuision of Biomedical and Enuironmental Re- search (DBER) of the U.S. Department of Energy (DOE), under contract EY-76-(2-03-0472 with the Uniuersity of California, Davis.

Volume 12, Number IO, October 1978 1167