Aerosol Particles in the Polluted and Global Atmosphere - Air Pollution and Global Warming: History, Science, and Solutions - page 112

particles o f volume − coagulate with particles of

volume minus the rate at which particles of volume

are lost due to coagulation with particles of all sizes.
The first integral in Equation 5.2 is multiplied by one-
half to eliminate double counting of production terms.

Example 5.1
Estimate the concentration of 0.2-

m-diameter
particles after 10 seconds of coagulation if
10 5 particles cm −3 of this size exist initially, no
other particles exist, and the coagulation rate
coefficient is 6

Figure 5.9. Schematic showing coagulation. When
two particles collide, they may coalesce to form one
large particle, thereby reducing the number
concentration but conserving the volume
concentration of particles.

10 −9 cm 3 particle −1 s −1 .Inthis
case, what is the resulting concentration of new
particles formed, assuming that they are formed
only from the 0.2-

×

Sulfuric acid-water-ammonia are more likely to homo-
geneously nucleate than sulfuric acid-water if all other
conditions are equal.

m-diameter particles and
none of the new particles is lost?

Solution
Because particles smaller than 0.2

5.3. Processes Affecting Particle Size
Once in the air, particles increase in size by coagula-
tion and growth. These processes are discussed in the
following subsections.

mindiame-
ter do not exist in this case, no new particles of
this size can form; thus, only the rightmost term
in Equation 5.2 needs to be considered. A sim-
ple discretization of the resulting equation gives
n t + h t n t n t ,where t is the initial time,

h is the time step, and the volume subscripts
were removed and replaced by time subscripts.
Based on the initial conditions provided, the loss
of 0.2-

= n t − h

5.3.1. Coagulation
Coagulation occurs when two particles collide and
stick together, or coalesce (Figure 5.9), reducing the
number concentration but conserving the volume of the
sum of all particles in the air. It can occur between two
small particles, a small and a large particle, or two large
particles.
Coagulation involves the simultaneous collision and
coalescence among many particles of different size.
Particles of a given volume

t n t n t =
600 particles cm −3 ,giving the new concentra-
tion of 0.2-

mparticles over h = 10 s is h

mparticles as 99,400 particles cm −3 .
Because each coagulation interaction resulted in
the loss of two smaller particles to produce one
larger particle, the number of new, larger parti-
cles formed must equal half the number of parti-
cles lost, thus 300 particles cm −3 .

(cm 3 per particle) are
produced by collisi on and coalescence of particles of
any smaller volume with particles of volume − .

Several important physical processes that drive par-
ticles to collide and stick together (coalesce), thereby
increasing the coagulation rate coefficient, are Brown-
ian motion, gravitational collection, and van der Waals
forces.
Brownian motion is the random movement of parti-
cles suspended in a fluid. Coagulation due to Brownian
motion is the process by which particles diffuse, col-
lide, and coalesce due to random motion. When two
particles collide due to Brownian motion, they may or
may not stick together, depending on the efficiency of
coalescence, which, in turn, depends on particle shape,
composition, and surface characteristics. Because the
kinetic energy of a small particle is small relative to
that of a large particle at a given temperature, the
likelihood that bounce-off occurs when small particles

However, particles of volume can collide and coalesce

with particles of any size to form larger particles. The
basic coagulation equation that describes the change
in the number concentration ( n , number of particles
per cubic centimeter of air) of particles of volume

due to these coagulation production and loss processes
is

∞

∂ n

∂

1
2

= − , n − n d − n , n d (5.2)

t

0 0

where is the coagulation rate coefficient (kernel) of

two colliding particles (cm 3 particle − 1 s − 1 ). Equation
5.2 states that the change in number concentration with
time of particles of volume equals the rate at which