The total surface area S, of the aerosol per cm3 of air is then

S, = n D2nN(Dp)dDp= ns{Dp)dDp (pm2cm"3) Jo Jo

(8.5)

and is equal to the area below the ns(Dp) curve in Figure 8.4. The aerosol volume distribution nv(Dp) can be defined as

nv(Dp)dDp — volume of particles per cm3 of air having diameters in the range Dp to (Dp + dDp)

and therefore

nv(Dp) =-D3pnN(Dp) (pm2cm~3)

(8.6)

The total aerosol volume per cm3 of air V, is

v,=- DlnN(Dp)dDp= nv{Dp)dDp (pm3cnr3) o Jo Jo

(8.7)

and is equal to the area below the ny(Dp) curve in Figure 8.4.

If the particles all have density pp (g cm-3) then the distribution of particle mass with respect to particle size, riM{Dp), is

M*>p) = ($)ny(Dp) = Q)DlnN(Dp) (pgpnT1 cuT3) (8.8)

where the factor 106 is needed to convert the units of density pp from g cm"3 to pg |im 3, and to maintain the units for >im(Dp) as pg pm 1 cm-3.

Because particle diameters in an aerosol population typically vary over several orders of magnitude, use of the distribution functions, nN(Dp), ns(Dp), tiy{Dp), and nM(Dp), is often inconvenient. For example, all the structure of the number distribution depicted in Figure 8.4 occurs in the region from a few nanometers to 0.3 pm diameter, a small part of the 0-10 pm range of interest. To circumvent this scale problem, the horizontal axis (abscissa) can be scaled in logarithmic intervals so that several orders of magnitude in Dp can be clearly seen (Figure 8.5). Plotting «a? (Dp) on semilog axes gives, however, a somewhat distorted picture of the aerosol distribution. The area below the curve no longer corresponds to the aerosol number concentration. For example, Figure 8.5 appears to suggest that more than 90% of the particles are in the smaller mode centered around 0.02 pm. In reality, the numbers of particles in the two modes are almost equal; this can be seen when the distributions are expressed as a function of the logarithm of diameter, which we discuss in the next section (Figure 8.6).

FIGURE 8.5 The same aerosol distribution as in Figure 8.4, plotted against the logarithm of the diameter.

Diameter, fj,m

FIGURE 8.5 The same aerosol distribution as in Figure 8.4, plotted against the logarithm of the diameter.

6000

4000

J* 2000

0 200

4000

J* 2000

FIGURE 8.6 The same aerosol distribution as in Figures 8.4 and 8.5 expressed as a function of log Dp and plotted against log Dp. Also shown are the surface and volume distributions. The areas below the three curves correspond to the total aerosol number, surface, and volume, respectively.

0.01

10.00

FIGURE 8.6 The same aerosol distribution as in Figures 8.4 and 8.5 expressed as a function of log Dp and plotted against log Dp. Also shown are the surface and volume distributions. The areas below the three curves correspond to the total aerosol number, surface, and volume, respectively.

Expressing the aerosol distributions as functions of In Dp or log Dp instead of Dp is often the most convenient way to represent the aerosol size distribution. Formally, we cannot take the logarithm of a dimensional quantity. Thus, when we write In Dp, we really mean In (Dp/1), where the "reference" particle diameter is 1 pm and is not explicitly indicated. We can therefore define the number distribution function n'v(lnD;,j as rfN(In Dp) d In Dp = number of particles per cm3 of air in the size range In Dp to (In Dp + d In Dp)

The units of neN(lnDp) are cm-3 since In Dp is dimensionless. The total number concentration of particles N, is

J — 00 |
(8.9) | ||

The limits of integration in (8.9) are from —oo to oo as the independent variable is In Dp. The surface area and volume distributions as functions of In Dp can be defined similarly to those with respect to Dp | |||

nes (In Dp) = 7i D2prfN (in Dp) (pm2cm-3) | |||

nev (\nDp) =^D3pneN (InDp) (pmW3) |
(8.10) | ||

with | |||

J —CO |
J —OO |
^ J - OO | |

J —OO |
(8.12) |

These aerosol distributions can also be expressed as functions of the base 10 logarithm log Dp, defining n°N(\ogDp), n°s(logDp), and «y(log Dp). Note that nN, neN, and n°N are different mathematical functions, and, for the same diameter Dp, they have different arguments, namely, Dp, In Dp, and log Dp. The expressions relating these functions will be derived in the next section.

Using the notation dN/dS/dV = the differential number/surface/volume of particles in the size range Dp to Dp + dDp we have dN = nN(Dp)dDp = neN (in Dp) d In Dp = nN(\ogDp)d\ogDp (8.13)

dS = ns(Dp)dDp = nes (in Dp) din Dp = n°s (log Dp)d log Dp (8.14)

dV = nv(Dp)dDp = nev(lnDp)dlnDp = n°v(logDp)dlogDp (8.15)

On the basis of that notation, the various size distributions are

ns(DP) = §• nes(\nDp) = ^ „S(logD,) = ^ (8.16)

and they represent the derivatives of the cumulative number/surface area/volume distributions N(DP)/S(DP)/V(Dp) with respect to Dp, In Dp, and log Dp, respectively.

8.1.4 Relating Size Distributions Based on Different Independent Variables

It is often necessary to relate a size distribution based on one independent variable, say, Dp, to one based on another independent variable, say, log Dp. Such a relation can be derived based on (8.13). The number of particles dN in an infinitesimal size range Dp to Dp + dDp is the same regardless of the expression used for the description of the size distribution function. Thus in the particular case of nN(Dp) and n°N(logDp)

Since d log Dp = d In Dp/2.303 = dDp/2303Dp, (8.17) becomes

»; (log Dp) = 2303DpnN(Dp) |
(8.18) | |

Similarly | ||

«5 (log Dp) = 2303Dpns(Dp) |
(8.19) | |

n°v(log Dp) = 2303Dpnv(Dp) |
(8.20) |

neN(\nDp) = DpnN(Dp) |
(8.21) |

nes (In Dp) = Dpns(Dp) |
(8.22) |

nev (In Dp) =Dpnv(Dp) |
(8.23) |

This procedure can be generalized to relate any two size distribution functions n(u) and n(v) where both u and v are related to Dp. The generalization of (8.17) is n(u) du = n(v) dv (8.24)

and dividing both sides by dDp

It is often convenient to summarize the features of an aerosol distribution using one or two of its properties (mean particle size, spread of distribution) than by using the full function nN(Dp). Growth of particles corresponds to a shifting of parts of the distribution to larger sizes or simply an increase of the mean particle size. These properties are called the moments of the distribution, and the two most often used are the mean and the variance.

Let us assume that we have a discrete distribution consisting of M groups of particles, with diameters D* and number concentrations Nk, k = 1,2,... ,M. The number concentration of particles is therefore

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