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Theoretical analysis of a method to measure size distributions of solid particles in water by aerosolization
Journal of Aerosol Science 83 (2015) 25–31
Contents lists available at ScienceDirect
Journal of Aerosol Science journal homepage: www.elsevier.com/locate/jaerosci
Theoretical analysis of a method to measure size distributions of solid particles in water by aerosolization Nobuhiro Moteki n, Tatsuhiro Mori Department of Earth and Planetary Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
a r t i c l e i n f o
abstract
Article history: Received 8 September 2014 Received in revised form 21 January 2015 Accepted 9 February 2015 Available online 17 February 2015
Reliable measurement of the size-resolved number concentration (size distribution) of solid particles dispersed in water or melted ice is of critical importance in many geoscientific observational studies. Because physical and chemical properties of particles can be measured more unambiguously and accurately in rarefied media (air) than in condensed media (water), particle measurement after aerosolization using a nebulizer is a significant method for the observation of solid particles dispersed in water. We propose a mathematical theory for estimating the original size distribution of solid particles in water from the measured size distribution of aerosolized particles. We assume that the size distribution of water droplets produced by a nebulizer is given. The size distribution of solid particles in water can be estimated by solving a system of nonlinear equations. The complexity in solving the equations increases rapidly with the computational resolution of particle size and the assumed maximum number of particles within each droplet. For such a system of equations, we found rigorous error bounds of a true solution using INTLAB, an interval arithmetic package. Our theoretical framework will be useful in many fields in geoscience as a fundamental scheme to quantify solid particles in water. In particular, an application of the proposed theoretical method is shown to be useful for the quantitative observations of the size distribution of black carbon particles in rainwater. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Colloid Size distribution Nebulizer Black carbon
1. Introduction Solid particles dispersed in water, snow, and ice on the earth's surface have many possible environmental and climatic impacts. Marine colloidal particles alter the transfer of solar radiation in seawater through scattering and absorption (Stramski et al., 2004) and may serve as nuclei for larger particles that control the gravitational sedimentation of geochemically important species (Wells & Goldberg, 1992). In groundwater, colloidal particles with size less than several micrometers can be a dominant agent for the transport of hazardous radionuclides and hydrophobic organic molecules across significant horizontal distances underground (McCarthy & Zachara, 1989). Light-absorbing particles (e.g., black carbon (BC)) deposited on snow reduce surface albedo and induce melting (Flanner et al., 2007). These climatic and environmental effects of small particles in water or ice strongly depend on their size, shape, chemical composition, and mixing state. Rainwater and snow contain aerosol particles incorporated through the nucleation of cloud and ice particles and coalescence with precipitating particles. Because the particle size of water-insoluble components does not change even
n
Corresponding author. E-mail address: [email protected] (N. Moteki).
http://dx.doi.org/10.1016/j.jaerosci.2015.02.002 0021-8502/& 2015 Elsevier Ltd. All rights reserved.
26
N. Moteki, T. Mori / Journal of Aerosol Science 83 (2015) 25–31
Nomenclature d D l L n n n0 n0
N N
diameter of particle, m diameter of droplet, m number of size bins for particle number of size bins for droplet number concentration of particles in water, m3 vector notation for size-resolved number concentration of particles in water, m 3 number concentration of particles in air, m 3 vector notation for size-resolved number concentration of particles in air, m 3
pij ðxi Þ qjk v V x Xk
λij
number concentration of droplet in air, m 3 vector notation for size-resolved number concentration of droplet in air, m 3 probability distribution for number of i-th particles xi in a j-th droplet probability of production of k-th particle from j-th droplet volume of a particle, m3 volume of a droplet, m3 number of particles in a droplet a set in l-dimensional space of non-negative integers defined by Eq. (4) mean number of i-th particles in a j-th droplet
after incorporation into liquid water, comparison between the size distributions of such particles in air and precipitating water is the basis for discussing the possible size-dependent efficiency of the wet removal of atmospheric aerosols. The combination of laser-diffraction tomography with fluorescence spectroscopy has been a major scheme for observing the size distribution and chemical characterization of solid particles in water (Kerker, 1983; Miroslaw & Georges, 2011). However, interpreting the microphysical properties of particles solely from these optical signals could sometimes yield ambiguous results, because the angular intensities of scattered light and fluorescence emission depend on the size, shape, chemical composition, and mixing state of individual particles (Wang et al., 1980). Recently, quantitative measurements of (black carbon) BC particles dispersed in water have become possible using a nebulizer and laser-induced incandescence (LII) technique (Ohata et al., 2011; Schwarz et al., 2013). To date, the LII technique, which requires aerosolization (i.e., extraction of the colloidal particles into air as aerosols) of particles dispersed in water before detection, is the only method available to measure the BC mass present in individual internally mixed multicomponent particles. In general, measurement techniques and their signal interpretations are simpler for particles suspended in rarefied media (vacuum or gas) than those dispersed in condensed media (liquid or solid). Therefore, for a number of geophysical, geochemical, and microbiological research applications, particle measurements after aerosolization is a fundamental scheme for the quantitative observations of solid particles in water. As illustrated in Fig. 1 (see also Fig. 10, Ohata et al., 2011), the aerosolization of solid particles in water using a nebulizer changes their size and number concentration if multiple particles are included within a droplet. The particles are agglomerated to a single larger particle after the evaporation of the droplet. Agglomerations enlarge size and reduce the number of particles observed in air compared to the original particles in water. The occurrence of agglomeration becomes negligible under the limit of dilute particle concentration, wherein individual droplets contain at most one particle. In this paper, we propose a mathematical theory to determine the original size distribution of solid particles in water by particle measurements after aerosolization using a nebulizer, assuming availability of experimental data of the size distribution of nebulized water droplets. This theoretical method is universally applicable to any observation of solid particles in water using a nebulizer.
2. Mathematical theory A schematic of the proposed theoretical method is shown in Fig. 2. Derivations of important formulae are given in this section. In our formulations, the size of particles and droplets is discretized into l and L bins, respectively. We denote a particle with i-th size bin simply as an i-th particle and a nebulized water droplet with j-th size bin as a j-th droplet.
Fig. 1. Aerosolization of solid particles dispersed in water using a nebulizer.
N. Moteki, T. Mori / Journal of Aerosol Science 83 (2015) 25–31
27
Mathematical descriptions of the size distributions of particles and droplets Droplet-size distribution with L bins
Particle-size distribution with l bins Number conc.
n1
n2
Volume
v1
v2
nl
N1
......
......
vl
......
Size binning v 1/2
v 3/2
v 5/2
V1
v l - 1/2
NL
N2
V2
VL
v l + 1/2
The probability of finding xi i-th particles in a j-th droplet: pij(xi)
Equation (2)
The probability of the production of a k-th particle from a j-th droplet: qij
Equation (3)
The number concentration of the k-th particles observed in air: nk' (k = 1, 2, ... , l)
Equation (5)
A system of nonlinear equations with respect to the unknown
Equation (6)
number concentration of particles in water nk (k =1,2, ... , l) Input
Input
Measured nk' (k =1,2, ... , l)
Nonlinear equation solver
Measured droplet number concentration Nj (j =1,2, ... , L)
Find nk (k =1,2, ... , l). Fig. 2. Calculation procedures in the proposed mathematical theory.
From the definition of the number concentration n as a mean number of particles found in a unit volume of water, the mean number of i-th particles included within a j-th droplet is as follows:
λij ¼ ni V j ;
ð1Þ
where ni is the number concentration of i-th particles in water and Vj is the volume of the j-th droplet. The number of particles x in each droplet produced by a nebulizer is a discrete random variable ðx ¼ 0; 1; 2; …Þ. Therefore, the probability of finding xi i-th particles within a j-th droplet is described as a Poisson distribution of parameter λij (Riley et al., 2006): pij ðxi Þ ¼
λxiji xi !
e λij :
ð2Þ
This is a probability distribution function with respect to a discrete variable xi ð ¼ 0; 1; 2; …Þ. As illustrated in Fig. 2, vi denotes the volume of i-th particle (i¼1,…, l). A k-th particle is assumed to be produced from a droplet only if the total volume of P particles in the droplet ( li ¼ 1 xi vi ) is between the lower and upper bounds of the volume corresponding to the k-th particle bin (vk 1=2 vk þ 1=2 , Fig. 2). Therefore, the probability of the production of a k-th particle from a j-th droplet is formulated as follows: " # l X qjk ¼ ∏ pij ðxi Þ ; ð3Þ ðx1 ;…;xl Þ A X k
i¼1
where the summand in brackets is the joint probability of the simultaneous inclusion of xi i-th particles ði ¼ 1; …; lÞ within a j-th droplet, and Xk is a set in the l-dimensional space of nonnegative integers. Xk is defined as follows: l X xi vi ovk þ 1=2 ; X k ¼ x1 ; …; xl A f0g [ Nvk 1=2 r ð4Þ i¼1
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N. Moteki, T. Mori / Journal of Aerosol Science 83 (2015) 25–31
where N denotes an entire set of natural numbers, and the condition on the right side of the vertical bar presents the allowed range of the total volume of particles within a droplet. Using qjk, the number concentration of k-th aerosolized particles n0k is formulated as follows: n0k ¼
L X
Nj qjk
ðk ¼ 1; …; lÞ;
ð5Þ
j¼1
where N j ðj ¼ 1; …; LÞ denotes the size distribution of droplets (Fig. 2) that are assumed to be known. The numerical evaluation of Eq. (5) is a forward problem that simulates the observed particle-size distribution in air n0 ðn01 ; …; n0l Þ for water samples of assumed particle-size distribution in water n ðn1 ; …; nl Þ. Although the forward problem is useful for test purposes, solving an inverse problem is of practical importance, i.e., the determination of unknown n of a water sample from the observed n0 . In the inverse problem, Eq. (5) should be considered as a system of l nonlinear equations with l unknowns n1 ; …; nl as follows: 8 f ðn ; …; nl ; n01 ; NÞ ¼ 0; > < 1 1 ⋮ ð6Þ > : f ðn ; …; n ; n0 ; NÞ ¼ 0: 1 l l l Here, N ðN 1 ; …; NL Þ and n0 in Eq. (6) are the parameters that are prescribed by measurements. From the evaluations of Eqs. (3) and (4), it is realized that nonlinear functions f 1 ; …; f l consist of the summations of considerable numbers of terms in the form p nx11 nx22 ⋯nxl l exp½ ðn1 V 1 þ n2 V 2 þ; ⋯; þnl V l Þ, and the number of terms increases explosively with the increase of l. Therefore, the functions f 1 ; …; f l are generally strongly nonlinear and have extremely long mathematical expressions. Our strategy for solving a system of nonlinear equations (6) will be explained in next section. 3. Computational aspects In this computation, before tackling Eq. (6), we neglect the contributions of terms containing xi greater than some threshold xmax . Here, a positive integer xmax is selected such that pij ðxi 4xmax Þ values for any i and j are approximately zero. The truncation of xi greater than xmax corresponds to the replacement of the expression x1 ; …; xl A f0; 1; …; 1g in Eq. (4) with x1 ; …; xl A f0; 1; …; xmax g. With this truncation approximation, Eq. (6) can be rewritten as follows: 8 f ðn ; …; nl ; n01 ; N; xmax Þ ¼ 0; > < 1 1 ⋮ ð7Þ > : f ðn ; …; n ; n0 ; N; x Þ ¼ 0: max l l 1 l A choice of some truncation threshold xmax can be validated by the precise agreement of the numerical solution with that of a repeated calculation with either xmax 1 or xmax þ 1. The smallest possible xmax is desired to suppress the length of the mathematical expression of each nonlinear function f 1 ; …; f l . From Eqs. (2)–(5), we computationally generated the strings of the mathematical expressions of f 1 ; …; f l that can be loaded by a nonlinear solver. The strong nonlinearity and huge length of the irreducible mathematical expressions of f 1 ; …; f l require particular consideration of the numerical methods used to solve a system of equations (7). Conventional methods for solving a system of nonlinear equations are commonly based on multidimensional Newton iterations starting from a point that is supposed to be near a solution. However, objective judgment of the validity of the numerical solutions has been impossible. Recently developed mathematical theories, computer assisted proof and self-validated computing (Moore et al., 2009), enable the determination of a narrow interval of floating point numbers enclosing a true solution of the problem. Proving the existence of a true solution within a given floating-point interval and iterative contractions of the interval enclosure of a solution is possible with the fixed-point theorem and interval arithmetic with outward rounding. We used the verifynlss.m function of an interval arithmetic library INTLAB (Rump, 1999, 2010) to compute a verified interval enclosure for a true solution of Eq. (7). In this study, for any n as a solution of the inverse problem, we presented only the midpoint of the interval because the relative width of the interval enclosure was always less than 10 13. 4. Results and discussion We show an example of an application of the proposed theory to observations of BC particles in rainwater. The rainwater sample was collected in Fukui, Japan (35.981N, 135.961E) during a rain event under the influence of the polluted outflow from the Asian continent. For this rainwater sample, measured BC mass concentration was 208 μg L 1. The rainwater sample was aerosolized using a pneumatic nebulizer (Marin-5, Teledyne CETAC Technologies, NE, USA). A single-particle soot photometer (Droplet Measurement Technologies, Inc., CO, USA) (Moteki & Kondo, 2010) was used to measure n0 of the aerosolized BC particles. The size distribution of water droplets produced by the Marin-5 was experimentally determined by measuring the size distribution of dried aerosol particles aerosolized from ðNH4 Þ2 SO4 aqueous solution with known molarities. In our experiment, size distribution of aerosolized particles dried through a Nafion tube (Perma Pure LLC) was measured by a scanning mobility particle sizer system consisting of a differential mobility analyzer (TSI model 3081) and a condensation
N. Moteki, T. Mori / Journal of Aerosol Science 83 (2015) 25–31
29
Fig. 3. Numerical solution of the inverse problem for the 20.8 μg L 1 sample. (a) Size distribution of BC particles in water n and that observed in air n0 and (b) concentration ratio n0 =n ðn01 =n1 ; …; n0l =nl Þ. Total volume concentration of droplets is shown in panel (b) as a blue line. In both panels, the black line totally overlaps with the red line. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 4. Numerical solution of the inverse problem for the 208 μg L 1 sample under the conditions shown in Fig. 3. In panel (a), the scale ratio of the left to right vertical axes was fixed to be the same as that shown in Fig. 3a. In both panels, the black line totally overlaps the red line. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 5. Numerical solution of the inverse problem for the 2080 μg L 1 sample under the same conditions shown in Fig. 3. In panel (a), the scale ratio of the left to right vertical axes was fixed to be the same as that shown in Fig. 3(a). (For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.)
particle counter (TSI model 3022A). The size distribution of nebulized water droplets was estimated from the measured aerosol particle-size distribution and the known concentration of the ðNH4 Þ2 SO4 in aqueous solutions. The estimated droplet size distribution was approximated by a log-normal distribution function with count median diameter of 3:5 μm and geometric standard deviation of 1:80. The total volume concentration of the water droplets in air was estimated as the volume weighted integral of the size distribution to be 9:38 10 5 cm3 =cm3 . We carried out sensitivity tests by changing the n0 to 1/10 and 10 times the rainwater sample (denoted as 20.8 and 2080 μg L 1 samples, respectively) to examine the possible changes of the results solely from the difference in particle number concentration. For each n0 data, the inverse problem was solved with computational parameters l¼7 and L¼8 with xmax ¼ 2 or 3.
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N. Moteki, T. Mori / Journal of Aerosol Science 83 (2015) 25–31
Fig. 6. Numerical solution of the forward problem under the same conditions shown in Fig. 4, but for doubled particle size resolution. In panel (a), the scale ratio of the left to right vertical axes was fixed to be the same as that shown in Fig. 4(a).
Figures 3–5 show calculated n for given n0 data (inverse problem, solving Eq. (7)). In each figure, panel (a) shows particlesize distributions n0 and n, whereas panel (b) shows the ratio n0 =n ðn01 =n1 ; …; n0l =nl Þ. The physical unit of n0 =n is equivalent to ðcm3 of water=cm3 of airÞ; the n0 =n converges to the total volume concentration of the nebulized water droplet (blue line in panel (b)) under the limit where the change of n0 =n by agglomeration can be ignored. As shown in panel (b), the discrepancies of n0 =n from the total volume concentration of the droplets were negligibly small for the 20.8 μg L 1 sample, indicating negligible agglomeration on aerosolization. Agglomeration increases with particle concentration and becomes appreciable for 208 and 2080 μg L 1 samples. Some discrepancies between results with xmax ¼ 2 and 3 were found for the 2080 μg L 1 sample, implying the need to repeat the calculation with xmax ¼ 4 to verify the correctness of the result with xmax ¼ 3. However, the computer used in this study (Apple MacBook Air, mid-2012 model, 8 GB RAM) does not have sufficient memory to load the extremely large mathematical expression of f 1 ; …; f l with xmax ¼ 4. Calculating n0 from assumed n (forward problem, simple numerical evaluation of Eq. (5)) is computationally less expensive than the inverse problem. Therefore, the forward problem would be the only choice for calculations with a higher particle size resolution (i.e., larger l) and denser particle concentration that require larger xmax . Under the experimental conditions shown in Fig. 4 (under l¼7 and xmax ¼ 3), we confirmed the agreement between the input of the inverse problem n0 and the output of the forward problem n0 with relative error less than 10 6 (forward problem n0 was calculated from the solution of the inverse problem n). Consistency between inverse and forward problems confirms the validity of the results of the forward problem obtained without the aid of verified computing. Figure 6 shows the results of the forward problem under the same conditions shown in Fig. 4, except that the particle size resolution had been doubled (l¼ 13). Input data n with l ¼13 was prepared by a spline interpolation of the solution of the inverse problem n for l ¼7. The detailed sizedependent changes of n0 =n due to agglomeration on aerosolization were successfully resolved by the calculation of the forward problem with l ¼13. This example illustrates the utility of the forward problem for the conditions with higher l and xmax that would be prohibitive for the inverse problems. Recently, Mori et al. (2014) observed that BC mass concentration in rainwater in Okinawa, Japan that is influenced by East Asian plumes can exceed 200 μg L 1. BC mass concentration in rainwater is expected to be much higher in polluted East Asian industrialized regions. As shown by our theoretical calculations, the size distribution of BC particles will be appreciably changed by aerosolization using the Marin-5 nebulizer for samples with BC mass concentration greater than 200 μg L 1. This suggests the practical importance of the proposed theory for the quantitative observations of the size distribution of BC in rainwater.
5. Conclusions We developed a mathematical theory to describe the relationship between the size distributions of solid particles in water before and after aerosolization under a priori knowledge of the size distribution of a water droplet produced by the nebulizer. The mathematical relationship can be solved as forward or inverse problems depending on the purpose and available computational resources. Solving the inverse problem enables determination of unknown size distributions of particles in water from the observed size distribution of aerosolized particles. The less computationally demanding forward problem is valuable for detailed theoretical tests of the nebulization system. Our example of BC size distribution in rainwater demonstrates the importance of considering the effects of agglomeration by nebulization if the number concentration of solid particles increases to a certain critical limit. The critical limit is changed depending on the size distribution of the droplets produced by a nebulizer and the error tolerance of size distribution measurements. For each experimental study using a nebulizer, use of the proposed theoretical method is recommended for reliable measurements of true size distribution of solid particles dispersed in water.
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Acknowledgments This study was funded by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (23221001) and the Global Environment Research Fund of the Ministry of the Environment, Japan (2-1403). We would like to thank S. Ohata and Y. Kondo for useful discussions. Program code for solving inverse and forward problems will be available on request to N. Moteki. References Flanner, M.G., Zender, C.S., Randerson, J.T., & Rasch, P.J. (2007). Present-day climate forcing and response from black carbon in snow. Journal of Geophysical Research: Atmospheres, 112, D11. Kerker, M. (1983). Elastic and inelastic light scattering in flow cytometry. Cytometry, 4, 1–10. McCarthy, J.E., & Zachara, J.M. (1989). Subsurface transport of contaminants. Environmental Science & Technology, 23, 496–502. Miroslaw, J., & Georges, F. (2011). Light Scattering by Particles in Water: Theoretical and Experimental Foundations. Academic Press: New York. Moore, R.E., Kearfott, R.B., & Cloud, M.J. (2009). Introduction to Interval Analysis. Siam: Philadelphia. Mori, T., Kondo, Y., Ohata, S., Moteki, N., Matsui, H., Oshima, N., & Iwasaki, A. (2014). Wet deposition of black carbon at a remote site in the East China Sea. Journal of Geophysical Research: Atmospheres, 119, 10485–10498. Moteki, N., & Kondo, Y. (2010). Dependence of laser-induced incandescence on physical properties of black carbon aerosols: Measurements and theoretical interpretation. Aerosol Science and Technology, 44, 663–675. Ohata, S., Moteki, N., & Kondo, Y. (2011). Evaluation of a method for measurement of the concentration and size distribution of black carbon particles suspended in rainwater. Aerosol Science and Technology, 45, 1326–1336. Riley, K.F., Hobson, M.P., & Bence, S.J. (2006). Mathematical Methods for Physics and Engineering 3rd edition). Cambridge University Press: Cambridge. Rump, S.M. (1999). Intlab—interval laboratory. In: T. Csendes (Ed.), Developments in reliable computing. Kluwer Academic Publishers: Dordrecht. Rump, S.M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287–449. Schwarz, J.P., Gao, R.S., Perring, A.E., Spackman, J.R., & Fahey, D.W. (2013). Black carbon aerosol size in snow. Scientific Reports, 3, 1356. Stramski, D., Boss, E., Bogucki, D., & Voss, K.J. (2004). The role of seawater constituents in light backscattering in the ocean. Progress in Oceanography, 61, 27–56. Wang, D.S., Kerker, M., & Chew, H.W. (1980). Raman and fluorescent scattering by molecules embedded in dielectric spheroids. Applied Optics, 19, 2315–2328. Wells, M.L., & Goldberg, E.D. (1992). Marine submicron particles. Marine Chemistry, 40, 5–18.
Contents lists available at ScienceDirect
Journal of Aerosol Science journal homepage: www.elsevier.com/locate/jaerosci
Theoretical analysis of a method to measure size distributions of solid particles in water by aerosolization Nobuhiro Moteki n, Tatsuhiro Mori Department of Earth and Planetary Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
a r t i c l e i n f o
abstract
Article history: Received 8 September 2014 Received in revised form 21 January 2015 Accepted 9 February 2015 Available online 17 February 2015
Reliable measurement of the size-resolved number concentration (size distribution) of solid particles dispersed in water or melted ice is of critical importance in many geoscientific observational studies. Because physical and chemical properties of particles can be measured more unambiguously and accurately in rarefied media (air) than in condensed media (water), particle measurement after aerosolization using a nebulizer is a significant method for the observation of solid particles dispersed in water. We propose a mathematical theory for estimating the original size distribution of solid particles in water from the measured size distribution of aerosolized particles. We assume that the size distribution of water droplets produced by a nebulizer is given. The size distribution of solid particles in water can be estimated by solving a system of nonlinear equations. The complexity in solving the equations increases rapidly with the computational resolution of particle size and the assumed maximum number of particles within each droplet. For such a system of equations, we found rigorous error bounds of a true solution using INTLAB, an interval arithmetic package. Our theoretical framework will be useful in many fields in geoscience as a fundamental scheme to quantify solid particles in water. In particular, an application of the proposed theoretical method is shown to be useful for the quantitative observations of the size distribution of black carbon particles in rainwater. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Colloid Size distribution Nebulizer Black carbon
1. Introduction Solid particles dispersed in water, snow, and ice on the earth's surface have many possible environmental and climatic impacts. Marine colloidal particles alter the transfer of solar radiation in seawater through scattering and absorption (Stramski et al., 2004) and may serve as nuclei for larger particles that control the gravitational sedimentation of geochemically important species (Wells & Goldberg, 1992). In groundwater, colloidal particles with size less than several micrometers can be a dominant agent for the transport of hazardous radionuclides and hydrophobic organic molecules across significant horizontal distances underground (McCarthy & Zachara, 1989). Light-absorbing particles (e.g., black carbon (BC)) deposited on snow reduce surface albedo and induce melting (Flanner et al., 2007). These climatic and environmental effects of small particles in water or ice strongly depend on their size, shape, chemical composition, and mixing state. Rainwater and snow contain aerosol particles incorporated through the nucleation of cloud and ice particles and coalescence with precipitating particles. Because the particle size of water-insoluble components does not change even
n
Corresponding author. E-mail address: [email protected] (N. Moteki).
http://dx.doi.org/10.1016/j.jaerosci.2015.02.002 0021-8502/& 2015 Elsevier Ltd. All rights reserved.
26
N. Moteki, T. Mori / Journal of Aerosol Science 83 (2015) 25–31
Nomenclature d D l L n n n0 n0
N N
diameter of particle, m diameter of droplet, m number of size bins for particle number of size bins for droplet number concentration of particles in water, m3 vector notation for size-resolved number concentration of particles in water, m 3 number concentration of particles in air, m 3 vector notation for size-resolved number concentration of particles in air, m 3
pij ðxi Þ qjk v V x Xk
λij
number concentration of droplet in air, m 3 vector notation for size-resolved number concentration of droplet in air, m 3 probability distribution for number of i-th particles xi in a j-th droplet probability of production of k-th particle from j-th droplet volume of a particle, m3 volume of a droplet, m3 number of particles in a droplet a set in l-dimensional space of non-negative integers defined by Eq. (4) mean number of i-th particles in a j-th droplet
after incorporation into liquid water, comparison between the size distributions of such particles in air and precipitating water is the basis for discussing the possible size-dependent efficiency of the wet removal of atmospheric aerosols. The combination of laser-diffraction tomography with fluorescence spectroscopy has been a major scheme for observing the size distribution and chemical characterization of solid particles in water (Kerker, 1983; Miroslaw & Georges, 2011). However, interpreting the microphysical properties of particles solely from these optical signals could sometimes yield ambiguous results, because the angular intensities of scattered light and fluorescence emission depend on the size, shape, chemical composition, and mixing state of individual particles (Wang et al., 1980). Recently, quantitative measurements of (black carbon) BC particles dispersed in water have become possible using a nebulizer and laser-induced incandescence (LII) technique (Ohata et al., 2011; Schwarz et al., 2013). To date, the LII technique, which requires aerosolization (i.e., extraction of the colloidal particles into air as aerosols) of particles dispersed in water before detection, is the only method available to measure the BC mass present in individual internally mixed multicomponent particles. In general, measurement techniques and their signal interpretations are simpler for particles suspended in rarefied media (vacuum or gas) than those dispersed in condensed media (liquid or solid). Therefore, for a number of geophysical, geochemical, and microbiological research applications, particle measurements after aerosolization is a fundamental scheme for the quantitative observations of solid particles in water. As illustrated in Fig. 1 (see also Fig. 10, Ohata et al., 2011), the aerosolization of solid particles in water using a nebulizer changes their size and number concentration if multiple particles are included within a droplet. The particles are agglomerated to a single larger particle after the evaporation of the droplet. Agglomerations enlarge size and reduce the number of particles observed in air compared to the original particles in water. The occurrence of agglomeration becomes negligible under the limit of dilute particle concentration, wherein individual droplets contain at most one particle. In this paper, we propose a mathematical theory to determine the original size distribution of solid particles in water by particle measurements after aerosolization using a nebulizer, assuming availability of experimental data of the size distribution of nebulized water droplets. This theoretical method is universally applicable to any observation of solid particles in water using a nebulizer.
2. Mathematical theory A schematic of the proposed theoretical method is shown in Fig. 2. Derivations of important formulae are given in this section. In our formulations, the size of particles and droplets is discretized into l and L bins, respectively. We denote a particle with i-th size bin simply as an i-th particle and a nebulized water droplet with j-th size bin as a j-th droplet.
Fig. 1. Aerosolization of solid particles dispersed in water using a nebulizer.
N. Moteki, T. Mori / Journal of Aerosol Science 83 (2015) 25–31
27
Mathematical descriptions of the size distributions of particles and droplets Droplet-size distribution with L bins
Particle-size distribution with l bins Number conc.
n1
n2
Volume
v1
v2
nl
N1
......
......
vl
......
Size binning v 1/2
v 3/2
v 5/2
V1
v l - 1/2
NL
N2
V2
VL
v l + 1/2
The probability of finding xi i-th particles in a j-th droplet: pij(xi)
Equation (2)
The probability of the production of a k-th particle from a j-th droplet: qij
Equation (3)
The number concentration of the k-th particles observed in air: nk' (k = 1, 2, ... , l)
Equation (5)
A system of nonlinear equations with respect to the unknown
Equation (6)
number concentration of particles in water nk (k =1,2, ... , l) Input
Input
Measured nk' (k =1,2, ... , l)
Nonlinear equation solver
Measured droplet number concentration Nj (j =1,2, ... , L)
Find nk (k =1,2, ... , l). Fig. 2. Calculation procedures in the proposed mathematical theory.
From the definition of the number concentration n as a mean number of particles found in a unit volume of water, the mean number of i-th particles included within a j-th droplet is as follows:
λij ¼ ni V j ;
ð1Þ
where ni is the number concentration of i-th particles in water and Vj is the volume of the j-th droplet. The number of particles x in each droplet produced by a nebulizer is a discrete random variable ðx ¼ 0; 1; 2; …Þ. Therefore, the probability of finding xi i-th particles within a j-th droplet is described as a Poisson distribution of parameter λij (Riley et al., 2006): pij ðxi Þ ¼
λxiji xi !
e λij :
ð2Þ
This is a probability distribution function with respect to a discrete variable xi ð ¼ 0; 1; 2; …Þ. As illustrated in Fig. 2, vi denotes the volume of i-th particle (i¼1,…, l). A k-th particle is assumed to be produced from a droplet only if the total volume of P particles in the droplet ( li ¼ 1 xi vi ) is between the lower and upper bounds of the volume corresponding to the k-th particle bin (vk 1=2 vk þ 1=2 , Fig. 2). Therefore, the probability of the production of a k-th particle from a j-th droplet is formulated as follows: " # l X qjk ¼ ∏ pij ðxi Þ ; ð3Þ ðx1 ;…;xl Þ A X k
i¼1
where the summand in brackets is the joint probability of the simultaneous inclusion of xi i-th particles ði ¼ 1; …; lÞ within a j-th droplet, and Xk is a set in the l-dimensional space of nonnegative integers. Xk is defined as follows: l X xi vi ovk þ 1=2 ; X k ¼ x1 ; …; xl A f0g [ Nvk 1=2 r ð4Þ i¼1
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where N denotes an entire set of natural numbers, and the condition on the right side of the vertical bar presents the allowed range of the total volume of particles within a droplet. Using qjk, the number concentration of k-th aerosolized particles n0k is formulated as follows: n0k ¼
L X
Nj qjk
ðk ¼ 1; …; lÞ;
ð5Þ
j¼1
where N j ðj ¼ 1; …; LÞ denotes the size distribution of droplets (Fig. 2) that are assumed to be known. The numerical evaluation of Eq. (5) is a forward problem that simulates the observed particle-size distribution in air n0 ðn01 ; …; n0l Þ for water samples of assumed particle-size distribution in water n ðn1 ; …; nl Þ. Although the forward problem is useful for test purposes, solving an inverse problem is of practical importance, i.e., the determination of unknown n of a water sample from the observed n0 . In the inverse problem, Eq. (5) should be considered as a system of l nonlinear equations with l unknowns n1 ; …; nl as follows: 8 f ðn ; …; nl ; n01 ; NÞ ¼ 0; > < 1 1 ⋮ ð6Þ > : f ðn ; …; n ; n0 ; NÞ ¼ 0: 1 l l l Here, N ðN 1 ; …; NL Þ and n0 in Eq. (6) are the parameters that are prescribed by measurements. From the evaluations of Eqs. (3) and (4), it is realized that nonlinear functions f 1 ; …; f l consist of the summations of considerable numbers of terms in the form p nx11 nx22 ⋯nxl l exp½ ðn1 V 1 þ n2 V 2 þ; ⋯; þnl V l Þ, and the number of terms increases explosively with the increase of l. Therefore, the functions f 1 ; …; f l are generally strongly nonlinear and have extremely long mathematical expressions. Our strategy for solving a system of nonlinear equations (6) will be explained in next section. 3. Computational aspects In this computation, before tackling Eq. (6), we neglect the contributions of terms containing xi greater than some threshold xmax . Here, a positive integer xmax is selected such that pij ðxi 4xmax Þ values for any i and j are approximately zero. The truncation of xi greater than xmax corresponds to the replacement of the expression x1 ; …; xl A f0; 1; …; 1g in Eq. (4) with x1 ; …; xl A f0; 1; …; xmax g. With this truncation approximation, Eq. (6) can be rewritten as follows: 8 f ðn ; …; nl ; n01 ; N; xmax Þ ¼ 0; > < 1 1 ⋮ ð7Þ > : f ðn ; …; n ; n0 ; N; x Þ ¼ 0: max l l 1 l A choice of some truncation threshold xmax can be validated by the precise agreement of the numerical solution with that of a repeated calculation with either xmax 1 or xmax þ 1. The smallest possible xmax is desired to suppress the length of the mathematical expression of each nonlinear function f 1 ; …; f l . From Eqs. (2)–(5), we computationally generated the strings of the mathematical expressions of f 1 ; …; f l that can be loaded by a nonlinear solver. The strong nonlinearity and huge length of the irreducible mathematical expressions of f 1 ; …; f l require particular consideration of the numerical methods used to solve a system of equations (7). Conventional methods for solving a system of nonlinear equations are commonly based on multidimensional Newton iterations starting from a point that is supposed to be near a solution. However, objective judgment of the validity of the numerical solutions has been impossible. Recently developed mathematical theories, computer assisted proof and self-validated computing (Moore et al., 2009), enable the determination of a narrow interval of floating point numbers enclosing a true solution of the problem. Proving the existence of a true solution within a given floating-point interval and iterative contractions of the interval enclosure of a solution is possible with the fixed-point theorem and interval arithmetic with outward rounding. We used the verifynlss.m function of an interval arithmetic library INTLAB (Rump, 1999, 2010) to compute a verified interval enclosure for a true solution of Eq. (7). In this study, for any n as a solution of the inverse problem, we presented only the midpoint of the interval because the relative width of the interval enclosure was always less than 10 13. 4. Results and discussion We show an example of an application of the proposed theory to observations of BC particles in rainwater. The rainwater sample was collected in Fukui, Japan (35.981N, 135.961E) during a rain event under the influence of the polluted outflow from the Asian continent. For this rainwater sample, measured BC mass concentration was 208 μg L 1. The rainwater sample was aerosolized using a pneumatic nebulizer (Marin-5, Teledyne CETAC Technologies, NE, USA). A single-particle soot photometer (Droplet Measurement Technologies, Inc., CO, USA) (Moteki & Kondo, 2010) was used to measure n0 of the aerosolized BC particles. The size distribution of water droplets produced by the Marin-5 was experimentally determined by measuring the size distribution of dried aerosol particles aerosolized from ðNH4 Þ2 SO4 aqueous solution with known molarities. In our experiment, size distribution of aerosolized particles dried through a Nafion tube (Perma Pure LLC) was measured by a scanning mobility particle sizer system consisting of a differential mobility analyzer (TSI model 3081) and a condensation
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Fig. 3. Numerical solution of the inverse problem for the 20.8 μg L 1 sample. (a) Size distribution of BC particles in water n and that observed in air n0 and (b) concentration ratio n0 =n ðn01 =n1 ; …; n0l =nl Þ. Total volume concentration of droplets is shown in panel (b) as a blue line. In both panels, the black line totally overlaps with the red line. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 4. Numerical solution of the inverse problem for the 208 μg L 1 sample under the conditions shown in Fig. 3. In panel (a), the scale ratio of the left to right vertical axes was fixed to be the same as that shown in Fig. 3a. In both panels, the black line totally overlaps the red line. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 5. Numerical solution of the inverse problem for the 2080 μg L 1 sample under the same conditions shown in Fig. 3. In panel (a), the scale ratio of the left to right vertical axes was fixed to be the same as that shown in Fig. 3(a). (For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.)
particle counter (TSI model 3022A). The size distribution of nebulized water droplets was estimated from the measured aerosol particle-size distribution and the known concentration of the ðNH4 Þ2 SO4 in aqueous solutions. The estimated droplet size distribution was approximated by a log-normal distribution function with count median diameter of 3:5 μm and geometric standard deviation of 1:80. The total volume concentration of the water droplets in air was estimated as the volume weighted integral of the size distribution to be 9:38 10 5 cm3 =cm3 . We carried out sensitivity tests by changing the n0 to 1/10 and 10 times the rainwater sample (denoted as 20.8 and 2080 μg L 1 samples, respectively) to examine the possible changes of the results solely from the difference in particle number concentration. For each n0 data, the inverse problem was solved with computational parameters l¼7 and L¼8 with xmax ¼ 2 or 3.
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Fig. 6. Numerical solution of the forward problem under the same conditions shown in Fig. 4, but for doubled particle size resolution. In panel (a), the scale ratio of the left to right vertical axes was fixed to be the same as that shown in Fig. 4(a).
Figures 3–5 show calculated n for given n0 data (inverse problem, solving Eq. (7)). In each figure, panel (a) shows particlesize distributions n0 and n, whereas panel (b) shows the ratio n0 =n ðn01 =n1 ; …; n0l =nl Þ. The physical unit of n0 =n is equivalent to ðcm3 of water=cm3 of airÞ; the n0 =n converges to the total volume concentration of the nebulized water droplet (blue line in panel (b)) under the limit where the change of n0 =n by agglomeration can be ignored. As shown in panel (b), the discrepancies of n0 =n from the total volume concentration of the droplets were negligibly small for the 20.8 μg L 1 sample, indicating negligible agglomeration on aerosolization. Agglomeration increases with particle concentration and becomes appreciable for 208 and 2080 μg L 1 samples. Some discrepancies between results with xmax ¼ 2 and 3 were found for the 2080 μg L 1 sample, implying the need to repeat the calculation with xmax ¼ 4 to verify the correctness of the result with xmax ¼ 3. However, the computer used in this study (Apple MacBook Air, mid-2012 model, 8 GB RAM) does not have sufficient memory to load the extremely large mathematical expression of f 1 ; …; f l with xmax ¼ 4. Calculating n0 from assumed n (forward problem, simple numerical evaluation of Eq. (5)) is computationally less expensive than the inverse problem. Therefore, the forward problem would be the only choice for calculations with a higher particle size resolution (i.e., larger l) and denser particle concentration that require larger xmax . Under the experimental conditions shown in Fig. 4 (under l¼7 and xmax ¼ 3), we confirmed the agreement between the input of the inverse problem n0 and the output of the forward problem n0 with relative error less than 10 6 (forward problem n0 was calculated from the solution of the inverse problem n). Consistency between inverse and forward problems confirms the validity of the results of the forward problem obtained without the aid of verified computing. Figure 6 shows the results of the forward problem under the same conditions shown in Fig. 4, except that the particle size resolution had been doubled (l¼ 13). Input data n with l ¼13 was prepared by a spline interpolation of the solution of the inverse problem n for l ¼7. The detailed sizedependent changes of n0 =n due to agglomeration on aerosolization were successfully resolved by the calculation of the forward problem with l ¼13. This example illustrates the utility of the forward problem for the conditions with higher l and xmax that would be prohibitive for the inverse problems. Recently, Mori et al. (2014) observed that BC mass concentration in rainwater in Okinawa, Japan that is influenced by East Asian plumes can exceed 200 μg L 1. BC mass concentration in rainwater is expected to be much higher in polluted East Asian industrialized regions. As shown by our theoretical calculations, the size distribution of BC particles will be appreciably changed by aerosolization using the Marin-5 nebulizer for samples with BC mass concentration greater than 200 μg L 1. This suggests the practical importance of the proposed theory for the quantitative observations of the size distribution of BC in rainwater.
5. Conclusions We developed a mathematical theory to describe the relationship between the size distributions of solid particles in water before and after aerosolization under a priori knowledge of the size distribution of a water droplet produced by the nebulizer. The mathematical relationship can be solved as forward or inverse problems depending on the purpose and available computational resources. Solving the inverse problem enables determination of unknown size distributions of particles in water from the observed size distribution of aerosolized particles. The less computationally demanding forward problem is valuable for detailed theoretical tests of the nebulization system. Our example of BC size distribution in rainwater demonstrates the importance of considering the effects of agglomeration by nebulization if the number concentration of solid particles increases to a certain critical limit. The critical limit is changed depending on the size distribution of the droplets produced by a nebulizer and the error tolerance of size distribution measurements. For each experimental study using a nebulizer, use of the proposed theoretical method is recommended for reliable measurements of true size distribution of solid particles dispersed in water.
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Acknowledgments This study was funded by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (23221001) and the Global Environment Research Fund of the Ministry of the Environment, Japan (2-1403). We would like to thank S. Ohata and Y. Kondo for useful discussions. Program code for solving inverse and forward problems will be available on request to N. Moteki. References Flanner, M.G., Zender, C.S., Randerson, J.T., & Rasch, P.J. (2007). Present-day climate forcing and response from black carbon in snow. Journal of Geophysical Research: Atmospheres, 112, D11. Kerker, M. (1983). Elastic and inelastic light scattering in flow cytometry. Cytometry, 4, 1–10. McCarthy, J.E., & Zachara, J.M. (1989). Subsurface transport of contaminants. Environmental Science & Technology, 23, 496–502. Miroslaw, J., & Georges, F. (2011). Light Scattering by Particles in Water: Theoretical and Experimental Foundations. Academic Press: New York. Moore, R.E., Kearfott, R.B., & Cloud, M.J. (2009). Introduction to Interval Analysis. Siam: Philadelphia. Mori, T., Kondo, Y., Ohata, S., Moteki, N., Matsui, H., Oshima, N., & Iwasaki, A. (2014). Wet deposition of black carbon at a remote site in the East China Sea. Journal of Geophysical Research: Atmospheres, 119, 10485–10498. Moteki, N., & Kondo, Y. (2010). Dependence of laser-induced incandescence on physical properties of black carbon aerosols: Measurements and theoretical interpretation. Aerosol Science and Technology, 44, 663–675. Ohata, S., Moteki, N., & Kondo, Y. (2011). Evaluation of a method for measurement of the concentration and size distribution of black carbon particles suspended in rainwater. Aerosol Science and Technology, 45, 1326–1336. Riley, K.F., Hobson, M.P., & Bence, S.J. (2006). Mathematical Methods for Physics and Engineering 3rd edition). Cambridge University Press: Cambridge. Rump, S.M. (1999). Intlab—interval laboratory. In: T. Csendes (Ed.), Developments in reliable computing. Kluwer Academic Publishers: Dordrecht. Rump, S.M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287–449. Schwarz, J.P., Gao, R.S., Perring, A.E., Spackman, J.R., & Fahey, D.W. (2013). Black carbon aerosol size in snow. Scientific Reports, 3, 1356. Stramski, D., Boss, E., Bogucki, D., & Voss, K.J. (2004). The role of seawater constituents in light backscattering in the ocean. Progress in Oceanography, 61, 27–56. Wang, D.S., Kerker, M., & Chew, H.W. (1980). Raman and fluorescent scattering by molecules embedded in dielectric spheroids. Applied Optics, 19, 2315–2328. Wells, M.L., & Goldberg, E.D. (1992). Marine submicron particles. Marine Chemistry, 40, 5–18.