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Anal. Chem. 1083,
Table VI. Chlorine Levels in Eluted Milk Fractions amt of C1, pg/mL fraction (mL) column 1 column 2 1 0.034 0.034 2 0.015 0.019 3 0.038 0.026 0.03 4 0.030 0.280 5 & 6* 0.294 7 0.064 0.066 8 11.0 0.3 9 149.0 23.6 10 234.0 216.0 11 170.0 38.0 12 3.52 lost 13 0.208 0.082 a Desalted milk, Table VII. Recovery of Heptachlor Spike Erom Blind Study p g of TOCl as fig of TOCl as heptachlor heptachlor spike added spike recovered 0.5 0.48 t 0.11 1.0 0.87 f 0.13 1.5 1.36 t 0.14 2.0 1.78 t 0.25 3.0 2.80 t 0.2 Na38C1spike. With NAA, human milk has been found experimentally to contain an average of 0.43 mg of Cl-/g; hence, less than 5 ng of C1- can be expected to carry over to the desalted milk fraction. This amount is considered within the blank values of the operating procedures. This experiment was run in duplicate on human milk, cows milk, Enfamil, and human serum. Results are summarized in Table V. When human milk was desalted according to the described procedure, the results shown in Table VI are typical. The fractions of interest contained 0.294 mg/mL chlorine and were preceded by a low chlorine fraction (0.03 pg) and followed by another low chlorine fraction (0.066 pg). Eluted fractions
55,94-98
containing NaCl appear later in fractions 10 and 11. A second column separation does not substantially improve the desired result further supporting the efficiency of the Bio-Gel column techniques. Figure 1 exhibits a y-ray spectra of neutron-irradiated human milk after desalting on Bio-Gel P-2. The heptachlor-spiked milk samples were submitted to the analyzing laboratory in a blind study to truly test the accuracy and reproducibility of the method. The method showed good reproducibility and accuracy as shown by the results gives in Table VII. TOCl from control milk (0.28 f 0.045 pg) was used to correct the results.
CONCLUSIONS This work describes a method which can be automated to a large extent and applied to large numbers of human body fluids for TOCl and TOBr analysis. These levels should be useful in continuing epidemological studies of the exposure of human populations to halogenated hydrocarbons in the environment. The method might also be useful as an ancillary method for determining fluid concentrations of halogenated hydrocarbons in toxicological studies with laboratory animals. ACKNOWLEDGMENT The authors thank Phillip Albro and his staff of the National Institute of Environmental Health Sciences for their help and assistance in this work. The experimental portion of this project was carried out by Science Applications, Inc., San Diego, CA. Registry No. p,p’-DDE, 72-55-9;heptachlor, 76-44-8. LITERATURE CITED (1) . . Biorseth. A,: Lunde, G.: Dvbina. . - E. Bull Environ. Contarn. Toxicol. 1977, 18 (9,581-587. (2) Neal, M. W.; Florlni, J. R. Anal. Blochem. 1973, 55, 328-330. (3) Uzlel, M.; Cohen, W. E. 8/0Chern. 8lophys. Acta 1865, 103,
(4) (5) (6) (7)
539-541 -. - . . . Ludkowitz, J. A.; Heurteblse, M. IAEA-SM-157/51, 437-447. Frltz, K.; Robertson, R. J. Radioanal. Chern. 1968, 1 , 463-473. Szilard, L.; Chalmers R. A. Nature (London) 1834, 132, 462. Gunnink, R.; Ruhter, W. D. GRPANL Lawrence Llvermoore Laboratory: UCRL-52917: Jan 1980.
RECEWED for review February 1,1982. Accepted July 19,1982.
Mathematical Model for Concentric Nebulizer Systems Anders Gustavsson Department of Analytical Chemistry, The Royal Institute of Technology, Fack,
A mathematlcal model for concentric nebullzer systems Is developed. The model Is usable for the calculation of the cutoff dlameter of the nebullzer system, the normal dlstrlbutlon parameters of the aerosol (the droplet dlstrlbutlon) generated by the nebulizer, the efflclency of the nebullzer system, and the aerosol concentration. The model also allows the optlmization of nebullzer systems. The mathematlcal model Is shown-by experlmentslo be In agreement with practlce.
There are very few articles of earlier date that describe theoretical work on nebulizers. But during the past years there has been an increased interest in nebulizers, nebulizer systems, and theory describing the way in which a nebulizer works.
S-100 44
Stockholm, Sweden
This increased interest has resulted in an increased number of papers during later years. Scientists working in spectroscopy have in the last years accepted the fact that it is impossible to investigate excitation sources (e.g., plasmas, flames) without a thorough knowledge of the nebulizer system. If this knowledge is missing it is not possible to take into consideration the influence of the nebulizer system on the measured properties of the excitation source. Thus, it w ill not be possible to separate the properties of the nebulizer system from the properties of the excitation source. Work on the measurement of aerosol dispersions has been published by Browner, Cresser, and Novak (1-4) and by Mohamed, Fry, and Wetzel (5). A paper on the effect of sample temperature in analytical flame spectroscopy has also been published by Browner and Cresser (6),where they have
0003-2700/83/0355-0094$01.50/00 1982 American Chemlcai Society
ANALYTICAL CHEMISTRY, VOL. 55, NO. 1, JANUARY 1983
Table I. A Comparison between Original and Recalculated d o ,Median, and Mean Values mean original values recalcd median from ref 7 and 8, values, exp(p* ), leXP(P* + (u*)*/2),
w
E.lm
Pm
15.6 17.7 19.8 19.9 22.6 25.0
15.7 17.7 19.9 19.9 22.4 25.2
13.7 13.7 13.8 13.8 13.8 13.9
_-
m
15.6 15.6 15.6 15.6 15.6 15.6
applied the results of Wukiyama and Tanasawa (7,8) to explain the small influence of the sample temperature on the signal. The investigation of the interference effects of aerosol ionic redistribution in analytical spectrometry (9) is important. A continued investigation of this phenomenon will probably explain some of the problems met when using nebulizer systems. From a theoretical point of view the cited form of the Nukiyama and Tanasawa equation (eq 8) in recent analytical literature deviates from the form used in this paper. The equation given in this paper is, however, in agreement with the form presented in the original paper (7). The deviations are related to the denominators of the equation, e.g., a missing p in the first term (1)and an improper square root in the second term (10-12). The main objective of this and earlier (13) work was to describe the combined process of aerosol generation, redistribution of the aerosol due to evaporation, and the separation of larger droplets in the spray chamber when nebulizing pure water. In ref 14 Smith and Browner have used another approach. They study the analyte transport efficiency but by doing this they have also included phenomena other than pure nebulization in their measuring object. There will be a need for more theoretical considerations to develop a model for analyte transport efficiency,which is the object of forthcoming work. The objectives of this work were as follows: (1)to sum up the contents of the Nukiyama and Tanasawa equation and to give a short survey of the work of Nukiyama and Tanasawa that relates to nebulizers; (2) to develop a mathematical model for nebulizer systems employing concentric nebulizers; (3) to compare the model with actual measurements on the real system presented in ref 13.
EXPERIMENTAL SECTION The equipment used for this study is listed in ref 13,but the spray chamber listed in Table I is not from Plasma-Therm but rather from Wiklunds. The spray chamber made by Wiklunds has a larger cutoff diameter than that of Plasma-Therm. Measurement Procedure. The measurement procedure for the liquid flows will be described in detail in a forthcoming paper but will be more closely diescribed here than in ref 13 because that text has been misunderstood. Before performing any measurements-at a specified pressure drop-the system has been equilibrated. That is, the system has been running for such a long time that a stable temperature and liquid flow pattern in the spray chamber have been obtained. Then-with the system running the entire time---the uptake and waste were measured by weighing. From the weights obtained the volume uptake and volume efficiency were calculated. The volume basis is chosen to facilitate the compariieon when nebulizing liquids of different density (e.g., Figures 4 and 5 in ref 13) and from the fact that the mass of analyte can he calculated directly from volumes if no analyte redistribution and/or severe evaporation occurs from the bulk of liquid in the spray chamber. The evaporation from the bulk of water will be of little consequence due to the rapid evaporation of small droplets in the aerosol saturating the gas stream. The measurement prlocedure is indirect as compared to the direct method employed by Smith and Rrowner (14). There are-in this case-five reasons for using an indirect method (1)
95
There is no analyte to be recovered due to the usage of pure water. (2) The time per measurement is small. (3)There will be no problems with water vapor in the ambient air. (4) Using a direct method would give severe problems with water evaporation. ( 5 ) The severe precision problem due to variability in the washout and in the atomic absorption measurement for indirect methods (14) is avoided because there is no analyte present.
THEORY A nebulizer system consists of two parts, the spray chamber and the nebulizer. The nebulizers dealt with throughout the remainder of this paper will be of the concentric type. The nebulizer characteristics used were defined earlier, in ref 13, but for the convenience of the reader they will be redefined here. Uptake: The volume of liquid (mL) aspirated by the nebulizer per minute. The input liquid head is at -5 to -7 cm. Gas flow: Flow of the nebulizing gas (L/min). Efficiency ( q ) : The part of the uptake nebulized (percent by volume). The nebulized part is measured as the difference between the uptake and the quantity of liquid waste, Le., including the water evaporated from the droplets. Aerosol concentration: Concentration of the aerosol measured as the volume of nebulized liquid (pL) per liter of nebulizing gas. Pressure drop: Drop of pressure across the nebulizer (bar). Before continuing with the theory it should be observed that the definition of, as well as the symbol for, the efficiency differs from that in ref 14 because of the different approaches chosen. The basic theory of nebulizer systems will be dealt with in three separate sections; namely, properties of log-normal distributions, theory of nebulizer, and theory of the nebulizer plus spray chamber. The reason for separating the last two theoretical items into two sections is to emphasize the characteristic properties of the nebulizer and the spray chamber separately. General Properties of log-Normal Distributions and the Validity of Using the Two-Parameter Model for Aerosols. When treating data-which are assumed to be distributed log-normally-we can use models with a number of parameters (15). The models most commonly used have two, three, or four parameters. Here the two-parameter model will be described, because a good fit to the values presented by Nukiyama and Tanasawa was obtained by using this model. If the distribution of a (positive) variate x is such that the distribution of the transformed variate y = In x is (exactly) normal (Gaussian), with mean p* and standard deviation u*, N(p*,u*),then the distribution of x is said to be log-normal. General properties of log-normal distributions are given in ref 15. In particular log-normal distributions possess moments of any order; the j t h moment is denoted by Aj, and from the properties of the moment-generating function of the normal distribution. The mean do and variance sd2 for the log-normal distribution are given by
+ (~*)~/2)
(2)
+ (u*)2)(exp((u*)2)- 1) = do2y2
(3)
do = exp(p*
s$ = exp(2p*
y = S d / d o = dexp((a*)2) - 1
(4)
is the coefficient of variation. For small values of u* we obviously have Sd/d(J
=
(r*
(5)
If two distributions have equal coefficients of variation they also have equal values of the parameter (r* and conversely. A characteristic property of log-normal distributions is that
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 1, JANUARY 1983
when converting them from number to mass distributions they will retain the log-normal behavior (ref 15, page 100). Hence, we can convert Nukiyama and Tanasawa’s number distributions into the corresponding mass distributions without losing the log-normal properties of the distribution. Another characteristic property of log-normal distributions (15)generated under equal circumstances, i.e., aerosols generated by the same nebulizer with different flows of gas and liquid, is the nearly constant value of the coefficient of variation sd/dO, where s d denotes the standard deviation of the log-normal distribution. As pointed out in ref 15, the log-normal distribution is well established in the domain of small-particle statistics. Distributions of small particles such as are found as the result of natural and mechanical processes are often very skew. The following test has been applied to check that the intuitively reasonable assumption of a log-normal distribution is valid for droplet diameters in an aerosol produced by concentric nebulizers. In ref 8 (report no. 1)six different measures of do were proposed for aerosol distributions. A numerical example of these measures for a particular aerosol is given in column 1 Table I (taken from report no. 1, Figure 12). The values in column 2 have the same moment as the corresponding values in column 1but they have been recalculated by using mean values of p* and u* determined from the original values in column 1 using eq 1. The recalculated median and mean values of the distribution are shown in Table I, columns 3 and 4. The p* and u* values for this recalculation have been obtained from the values taken as pairs from column 1 (the last value having been paired with the f i s t one). The similarity within the two columns three and four, respectively, and the similarity in pairs between the columns one and two shows that the two-parameter log-normal model satisfactorily describes the distribution of droplet diameters in an aerosol. Theory of the Nebulizer. The theory presented in this section w ill be based on the work of Nukiyama and Tanasawa (7,8).The velocities of the gas (u,) and the liquid (uI)at the nozzle face are calculated from the following expressions: ug
=
vi
= QdAi
Qg/Ag
(6) (7)
where Q, and Q1are the volume flows and A, and A, are the smallest areas at the nozzle face for the gas and the liquid flow, respectively. As seen from eq 6 and 7 there is no correction for stream contraction or velocity reduction. These corrections and the fluid dynamics of nebulizers will be dealt with in a future paper. The basic idea of Nukiyama and Tanasawa was to describe the (population) mean droplet diameter (do)of the droplets in the aerosol as a function of the different velocities, flows, and constants of the gas and the liquid. It should be emphasized that Nukiyama and Tanasawa treated the aerosol distributions as distributions by number and not by mass as in recent works (1-4). As can be seen from the statistical treatment in the previous section, from the theoretical treatment of small particle distributions in ref 15, from Figure 12 in report no. 1 (8),and from all of the figures in ref 16 that relate to droplet distributions, an aerosol distribution is of the logarithmic-normal (log-normal) type, i.e., by plotting the diameter on a logarithmic scale-in the distribution p l o t w e will get a normal distribution. Instead of treating the distributions as log-normal distributions Nukiyama and Tanasawa treated them as skew distributions only. The six different measures for do proposed by Nukiyama and Tanasawa gave different values of dodue to this fact. Of the six measures Nukiyama and Tanasawa chose the area-volume measure as being the one giving large droplets a high weight in the de-
ci
4
Diamefer ,p m
Figure 1. The typical appearanceof a log-normal distributionof droplet diameters in an aerosol.
termination of do. For Nukiyama and Tanasawa this was a good choice since they were studying carburetor theory, but for nebulizer theory better choices can be made. The continued work of Nukiyama and Tanasawa showed that do did not depend on the shape or size of the liquid and the gas nozzles or on their relative axial positions (ref 7, report no. 2). Three different types of air nozzles were tried, a convergent, a straight-bored, and a knife-edge nozzle. If Q, was increased, do decreased but only to the limit where Qg/Q1 = 5000. Nukiyama and Tanasawa concluded that do was a function of the velocity difference ( u p- ul) and Q1/Qg. The final equation presented by Nukiyama and Tanasawa for do (pm) was
Here, p is the density (g/cm3), u is the surface tension (dyn/cm), p is the coefficient of viscosity ((dyn s)/cmz),and u is the velocity difference ug - u1 (m/s). The experimental ranges for the constants were 0.8 < p > 1.2, 30 < u < 13 and 0.01 < p < 0.3. p , u, and p are of course related to the liquid. The Nukiyama and Tanasawa equation is strictly valid only when using air as the gas, but using another gas having nearly the same velocity of sound (e.g., argon) will give rise to a negligible error only. Equation 8 is in agreement with Nukiyama and Tanasawa’s experimental findings, but it should be observed that the two members of the equation are not dimensionally equal. This does not matter when the equation is used within its limitations but it is not attractive from a scientific point of view. Before treating the nebulizer and the spray chamber as one unit, one should pay attention to the fact that the nebulizer is a device producing a log-normal droplet distribution (an aerosol) of mean diameter do. Theory of the Nebulizer Plus Spray Chamber. The aerosol is forced to pass through a spray chamber. We can look upon this process as if the aerosol were passing through a theoretical filter having a cutoff diameter (d,). Droplets with a diameter I d , will pass on unaffected and droplets larger than d, will be retained (the waste). A picture of the selection process is given in Figure 1. The shaded area represents the droplets passing through the spray chamber. The selection process illustrated is of course a simplified representation of what actually occurs because the final aerosol will have droplets larger than d,. This is due to the nontheoretical behavior of the filter (spray chamber). The efficiency of the nebulizer system, i.e., the probability (proportion) of droplets having diameters I d , , can be calculated from the expression @((lnd, - In d o ) / u * ) (9) where denotes the standardized normal distribution function and u* denotes the standard deviation of the normal distri-
ANALYTICAL CHEMISTRY, VOL. 55, NO. 1, JANUARY 1983
97
1 74
Flgure 2. The estimation crf u* and d , using a normal probabilf paper. The horizontal line (- -) denotes the 16% level.
.
bution corresponding to the log-normal distribution. To s u m up this section, the efficiency of a nebulizer system can be calculated if do, d,, and s d or u* are known or if do is estimated with eq 8 and d, and Sd/do or u* are determined in some other way (the relation between Sd/do and u* is given in a previous section). The theory of the spray chamber is of course also valid for aerosols produced by other types of nebulizers. RESULTS AND DISCUSSION The theory presented will be used to elucidate three different problems: (1)the determination of d, and u*; (2) the calculation of the efficiency and the aerosol concentration for a particular nebulizer system; (3) optimization of nebulizer systems. Experimental conditions and data for this theoretical treatment are taken from ref 13. The determination of d, is direct and closely related to determining u* (the determination of d, according to ref 1.-4 and 17 is an alternative procedure). The nebulizer characteristics for a number of pressure drops are measured for a particular nebulizer system. The efficiencies are calculated from these nebulizer characteristics and the corresponding dovalues from the nebulizer characteristics and the Nukiyama and Tanasawa equation. The efficiencies and their corresponding dovalues are plotted on a normal probability paper, and the points should fit to a straight line, A, as exemplified in Figure 2. Next, the line A is extended until it intersects the 50% level, which occurs at In (d,). u* is obtained as the distance B at the 16% level. These percentage levels are clear from ref 15 and the properties of normal probability papers. The d, value for this particular nebulizer system (13)is = 4.5 pm, which is in accordance with published work on aerosol measurements (ref 1, Figure 6) using a similar spray chamber. It could be somewhat surprising that the present result is in accordance with ref 1, as Nukiyama and Tanasawa used number distributions, while the other authors used mass distributions as a base for the calculations. However, Nukiyama and Tanasawa used a measure of do which emphasized the large droplets. Conversion of the number distributions given in ref 8 to mass distributions w i l l yield mass distributions having a mean droplet. diameter close to do. The calculation of the efficiency and the aerosol concentration for a particular nebulizer system starts with the determination of the d, v
