Encyclopedia of Environmental Science and Engineering, Volume I and II

The log-normal distribution is particularly useful for rep-

resenting aerosols because it does not allow negative particle

sizes. The log-normal distribution function is obtained by

substituting ln D p and ln  g for D p and s in Eq. (4),

.

fD

DD

p

g

p p

g

ln

ln ln

().

()

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟



1 

2 2

2

2

π s s

exp

ln ln

(8)

The log-normal distribution has the following cumulative

distribution,

F

DD

D

g

pg

g

D

p

p



1 

2 2

2

0 2

psln s

exp

ln ln

ln

.

⎛ ()

⎝

⎜

⎜

⎞

⎠

⎟

⎟

∫ dln() (9)

The geometric mean diameter D g , and the geometric standard

deviation s g , are determined from particle count data by

ln ln

ln ln ln.

DnDN

nD DN

gipi

gipig







∑

∑

()

⎡⎣⎢ ()⎤⎦⎥

/

s /

2 12 (10)

Figure 1 shows the log-normal size distribution for par-

ticles having D g  1 m m and s g  2.0 on a log-probability

graph, on which a log-normal size distribution is a straight

line. The particle size at the 50 percent point of the cumu-

lative axis is the geometric mean diameter D g or number

median diameter, NMD. The geometric standard deviation

is obtained from two points as follows:

sg

p

p

p

po

DF

DF

DF

DF













at

at

at

at

84 13

50

50

15 7

.%

%

%

.%

.

The rapid graphical determination of the geometric

mean diameter D g as well as the standard deviation s g is a

major advantage of the log-normal distribution. It should

be emphasized that the size distribution on a number basis

shown by the solid line in Figure 1 differs significantly

from that on a mass basis, shown by the dashed line in the

same figure. The conversion from number median diameter

(NMD) to mass median diameter (MMD) for a log-normal

distribution is given by

ln(MMD)  ln(NMD) 
 3(ln s g )^2. (11)

If many particles having similar shape are measured on the

basis of one of the characteristic diameters defined in Table 1,

a variety of average particle diameters can be calculated as

shown in Table 2. The comparison among these diameters is

shown in Figure 1 for a log-normal size distribution. Each

average diameter can be easily calculated from s g and NMD

(or MMD).

Figure 2 indicates approximately the major sources of

atmospheric aerosols and their surface area distributions.

There tends to be a minimum in the size distribution of

atmospheric particles around 1 m m, separating on one hand

the coarse particles generated by storms, oceans and volca-

noes and on the other hand the fine particles generated by

fires, combustion and atmospheric chemistry. The commi-

nution processes generate particles in the range above 1 m m

and molecular processes lead to submicron particles.

PARTICLE DYNAMICS AND PROPERTIES

Typical size-dependent dynamic properties of particles sus-

pended in a gas are shown in Figure 3 together with defining

equations (Seinfeld, 1986). The solid lines are those at atmo-

spheric pressure and the one-point dashed lines are at low

pressure. The curves appearing in the figure and the related

particle properties are briefly explained below.

Motion of Large Particles

A single spherical particle of diameter D p with a velocity u in

air of density r f experiences the following drag force,

F d  C D A p ( r f u^2 /2) (12)

10 –3 10 –2 10 –1

10 –1

100

100

101

102

103

104

105

101 102 103

Dp (mm)

FOREST FIRE

PLUMES

INTENSE

SMOG

HEAVY

AUTO

TRAFFIC

VOLCANIC PLUMES

DUST STORMS

SAND

STORMS

INDUSTRY

TYPICAL URBAN

POLLUTION

SEA SALT

SOUTH ATLANTIC

BACKGROUND

NORTH ATLANTIC

BACKGROUND

CONTINENTAL

SURFACE AREA DISTRIBUTION, BACKGROUND

∆S/

∆log D

(p

mm

2 cm

-^3

)

FIGURE 2 Surface area distributions of natural and anthropo-

genic aerosols.

18 AEROSOLS

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