The mean particle diameter, Dp, of the population is

The variance a2, a measure of the spread of the distribution around the mean diameter Dp, is defined by a2 = Ek=iMDk - Dp) = i^Nk{Dk _ Dpf {m

A value of a2 equal to zero would mean that every one of the particles in the distribution has precisely diameter Dp. An increasing a2 indicates that the spread of the distribution around the mean diameter Dp is increasing.

We will usually deal with aerosol distributions in continuous form. Given the number distribution nN(Dp), (8.27) and (8.28) can be written in continuous form to define the mean particle diameter of the distribution by

$~DpnN{Dp)dDp 1 °> ~ j-nN{Dp)dDp ~ N, L DpHN(8'29)

and the variance of the distribution by a "-rnN(Dp)dDp--NJ0 {Dp ~ Dp) nN{Po)dD^ ^

Table 8.2 presents a number of other mean values that are often used in characterizing an aerosol size distribution.

TABLE 8.2 Mean Values Often Used in Characterizing an Aerosol Size Distribution

Property Defining Relation Description

Number mean diameter Dp

Median diameter

^med

Mean surface area S

Mean volume V

Surface area mean diameter D$

Volume mean diameter Dv

Surface area median diameter DSm

Volume median diameter DVm

Mode diameter

^mode

î>p = bSDPn»(Dp)dDp J®"0 tin(Pp)dDp = \Nt

V dDP )Dmit

Average diameter of the population

Diameter below which one-half the particles lie and above which one-half the particles lie Average surface area of the population

Diameter of the particle whose surface area equals the mean surface area of the population Diameter of the particle whose volume equals the mean volume of the population Diameter below which one-half the particle surface area lies and above which one-half the particle surface area lies Diameter below which one-half the particle volume lies and above which one-half the particle volume lies Local maximum of the number distribution

A measured aerosol size distribution can be reported as a table of the distribution values for dozens of diameters. For many applications carrying around hundreds or thousands of aerosol distribution values is awkward. In these cases it is often convenient to use a relatively simple mathematical function to describe the atmospheric aerosol distribution. These functions are semiempirical in nature and have been chosen because they match well observed shapes of ambient distributions. Of the various mathematical functions that have been proposed, the lognormal distribution (Aitchison and Brown 1957) often provides a good fit and is regularly used in atmospheric applications. A series of other distributions are discussed in the next section.

The normal distribution for a quantity u defined from — oc < u < oo is given by where u is the mean of the distribution, a^ is the variance and

The normal distribution has the characteristic bell shape, with a maximum at u. The standard deviation, crH, quantifies the width of the distribution, and 68% of the area below the curve is in the range u ± a„.

A quantity u is lognormally distributed if its logarithm is normally distributed. Either the natural (In u) or the base 10 logarithm (log u) can be used, but since the former is more common, we will express our results in terms of In Dp. An aerosol population is therefore log-normally distributed if u = In Dp satisfies (8.31), or where N, is the total aerosol number concentration, and Dpg and ag are for the time being the two parameters of the distribution. Shortly we will discuss the physical significance of these parameters. The distribution n^{Dp) is often used instead of n^(ln Dp). Combining (8.21) with (8.33)

A lognormal aerosol distribution with Dpg = 0.8 pm and <JX = 1.5 is depicted in Figure 8.7.

FIGURE 8.7 Aerosol distribution functions, iin(Dp), n°N(logDp) and neN{\aDp) for a lognormally distributed aerosol distribution Dpg = 0.8 |im and a,, = 1.5 versus log D„. Even if all three functions describe the same aerosol population, they differ from each other because they use a different independent variable. The aerosol number is the area below the «'v(log Dp) curve.

FIGURE 8.7 Aerosol distribution functions, iin(Dp), n°N(logDp) and neN{\aDp) for a lognormally distributed aerosol distribution Dpg = 0.8 |im and a,, = 1.5 versus log D„. Even if all three functions describe the same aerosol population, they differ from each other because they use a different independent variable. The aerosol number is the area below the «'v(log Dp) curve.

We now wish to examine the physical significance of the two parameters Dpg and <yg. To do so we will use the cumulative size distribution N(DP).

If the aerosol distribution is lognormal, nN(Dp) is given by (8.34) and therefore

To evaluate this integral we let

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