The Size Distribution Function

An aerosol particle can be considered to consist of an integer number k of molecules or monomers. The smallest aerosol particle could be defined in principle as that containing two molecules. The aerosol distribution could then be characterized by the number concentration of each cluster, that is, by Nk, the concentration (per cm3 of air) of particles containing k molecules. Although rigorously correct, this discrete method of characterizing the aerosol distribution cannot be used in practice because of the large number of molecules that make up even the smallest aerosol particles. For example, a particle with a diameter of 0.01 pm contains approximately 104 molecules and one with a diameter of 1 pm, around 1010. 

Atmospheric Chemistry and Physics From Air Pollution to Climate Change, Second Edition, by John H. Seinfeld and Spyros N. Pandis. Copyright 2006 John Wiley Sons, Inc. 

TABLE 8.1 Example of Segregated Aerosol Size Information 

The size distribution of a particle population can also be described by using its cumulative distribution. The cumulative distribution value for a size section is defined as the concentration of particles that are smaller than or equal to this size range. For example, for the distribution of Table 8.1, the value of the cumulative distribution for the 0.03-0.04 pm size range indicates that there are 350 particles cm-3 that are smaller than 

FIGURE 8.1 Histogram of aerosol particle number concentrations versus the size range for the distribution of Table 8.1. The diameter range 0-0.2 pm for the same distribution is shown in the inset. 

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Here, Ari) is the size distribution function of the initial solid reagents... 

In order to characterize quantitatively the polydisperse morphology, the shape and the size distribution functions are constructed. The size distribution function gives the probability to find a droplet of a given area (or volume), while the shape distribution function specified the probability to find a droplet of given compactness. The separation of the disconnected objects has to be performed in order to collect the data for such statistics. It is sometimes convenient to use the quantity v1/3 = [Kiropiet/ ]1 3 as a dimensionless measure of the droplet size. Each droplet itself can be further analyzed by calculating the mass center and principal inertia momenta from the scalar field distribution inside the droplet [110]. These data describe the droplet anisotropy. 

A knowledge of the size distribution function of the radioactive debris and the specific activity of individual fission product chains as a function of particle size suffice to define many important radiological properties of the land-surface nuclear explosion. If is the function of a radionuclide or fission mass chain distributed between particle sizes Di and D2, then... 

The size distribution function f(D) pertaining to the event as a whole has variously been taken as a log-normal or as a power law distri-... 

In the majority of cases when one deals with nanosize superparamagnetic grains, polydispersity seems to be an inherent feature. The independent measurement of the size distribution function, such as by electron micrography, is a painstaking and rare opportunity. Besides, even when it is done, from the statistical viewpoint a set of available measurements (103 — 104 grains) for the particle number concentration even as small as 1010 — 1018, that is, 0.01% by volume at the particle size 10 nm, is far from being statistically representative. 

In contrast to the M(H) curves above TB, that are rather insensitive to the size distribution function, the high-temperature equilibrium susceptibility is quite sensitive to the contribution due to the largest particles. Using this fact the fit of l/y vs T (see Fig. 3 inset) yields a rather accurate estimate of the distribution width cr. This is favorable since the paramagnetic contribution can be neglected in this range. The values of p) and cr obtained form the... 

The moments of the size distribution function are useful parameters. These have the form ... 

The shape of the size distribution function for aerosol particles is often broad enough that distinct parts of the function make dominant contributions to various moments. This concept is useful for certain kinds of practical approximations. In the case of atomospheric aerosols the number distribution is heavily influenced by the radius range of 0.005-0.1 /xm, but the surface area and volume fraction, respectively, are dominated by the range 0.1-1.0 fxm and larger. The shape of the size distribution is often fit to a logarithmic-normal form. Other common forms are exponential or power law decrease with increasing size. 

Furthermore, the size distribution function can be retrieved by integration over all chemical species. 

If one focuses on the particle size distribution function as a central framework for describing aerosols, one can conveniently classify the measurement instruments according to the properties of the size distribution function. Organization of instrumentation gives perspective on the ideal requirements as contrasted with the practical limits imposed by current technology. An idealized hierarchy was suggested by S. K. Friedlander in 1977. As an ideal, the modern aerosol analyzer gives a continuous... 

The decomposition of the standard chemical potentials per amphiphile of the aggregates permits the size distribution function to be written as a product of exp [—[iig/hT)], which is a decreasing function of g, and another factor containing the size independent quantity . If is small, the size distribution is a monotonic decreasing function of g. If, however, is sufficiently large compared to unity, the size distribution can have a maximum. Therefore, a critical value of exists, separating the two kinds of behavior, To compute this critical value as well as the corresponding critical concentration of amphiphile, explicit expressions for the terms of the standard chemical potentials are needed. 

The size distribution of micellar aggregates Ng/F is plotted against the aggregation number, g, for an amphiphile with an octyl hydrocarbon tail and for a 2 X 1G4 cal A2t molsb (Figure 1). Equation 19 leads to wjt = 3.88. For < crit, the size distribution is a monotonic decreasing function of g. At = CT t, the size distribution function has ah inflection point. At > mt, the size distribution function has two extrema. It can be seen that if increases both the number and the average size of the micellar aggregates increase. 

There is some analogy between the critical concentration as defined here and the critical temperature predicted by the van der Waals equation of state, since each of them separates two kinds of behavior of the size distribution function and pressure-volume relationship, respectively. 

The introduction of this correction makes the algebra leading to the equations for gCTn and c,a more complex. For this case, the value of eJjt can be computed only by numerical methods. Figure 2 represents the size distribution function for 4 = 3 A and a 8 X 104 cal A2/mol for an amphiphile with an octyl hydrocarbon tail. 

Several parameters can be used to express the mean size of the milk fat globules. These parameters are derived from the so-called moments of the size distribution function the th moment of the distribution function (Sn) is equal to ... 

The normalization of the size distribution function P ( g) finally yields... 

Adopting the approach developed above for the char particles combustion, the size distribution function of limestone particles as a result of sulfation reaction in the overflow stream which is the same as in the bed is given by. 

In the quadrature method of moments (QMOM) developed by McGraw [131], for the description of sulfuric acid-water aerosol dynamics (growth), a certain type of quadrature function approximations are introduced to approximate the evolution of the integrals determining the moments. Marchisio et al [122, 123] extended the QMOM for the application to aggregation-breakage processes. For the solution of crystallization and precipitation kernels the size distribution function is expressed using an expansion in delta functions [122, 123] ... 

Substituting the self-similar form for the size distribution function, (5.23), we obtain... 

The integral in (5,24a) is a constant that depends on the form of the size distribution function, For the special case of coagulating, coalescing aerosols compjosed of spherical particles, the integral is 2.0 (Chapter 7) and... 

The coelficients of are momenls of the size distribution function known as cumulants. In practice, only the first two cumuJants can be accurately determined from the experimental data ... 

For a poiydisperse aerosol, the number of particles deposited up to any point in the system can be calculated from the theory for monodisperse aerosols and then integrating over the initial. size distribution, which is the quantity sought- The experimental measure ments made with the condensation nuclei counter gives the number concentration of the poiydisperse aerosol as a function of the distance from the inlet to the diffusion battery. The recovery of the size distribution function from the measured decay In particle concentration can be accomplished in an approximate way. Various numerical schemes based on plausible approximations have been developed to accomplish the inversion (Cheng, 1993). The lower detection limit for the diffusion battery is 2 to 5 nm. Systems are not difficult to build for specific applications or can be purchased commercially. 

It is desired to relate measured values of the light scattered by the atmo.spheric aerosol to the contributions of different ranges of the size distribution function. Discuss the components of a measurement system capable of providing the necessary information. Discuss a.ssumptions that must be made in carrying out the relevant calculations. 

A method of solving many coagulation anti agglomeration problems (Chapter 8) has been developed based on the use of a similarity transformation for the size distribution function (Swift and Fricdlander, 1964 Friedlander and Wang. 1966). Solutions found in this way are asymptotic forms approached after long times, and they are independent of the initial size distribution. Closed-form solutions for the upper and lower ends of the distribution can sometime.s be obtained in this way, and numerical methods can be used to match the solutions for intermediate-size particles. Alternatively, Monte Carlo and discrete sectional methods have been used to find solutions. 

To evaluate a it is necessary to solve for the SPD, which depends on Df. Values of a for the free molecule regime vary little with Df in the range 2 to 3 as shown in Table 8.1. along with the 1/0/ moment of the size distribution function. Mi/Of For Df 3, ctsa function of the size of the primary particle, ctpo. 

Aerosol growth laws are expressions for the rate of change in particle size as a function of particle size and the appropriate chemical and physical properties of the system. Such expressions are necessary for the calculation of changes in the size distribution function with time as shown in thi.s and the next chapter. In this section, transport-limited growth laws based on the previous section are discussed first followed by growth laws determined by aerosol phase chemical reactions. 

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