Encyclopedia of Environmental Science and Engineering, Volume I and II

AEROSOLS 21

from t  1s in Eq. (3.4). The intersection of the curves  x

and v t lies at around 0.5 m m at atmospheric pressure. If one

observes the settling velocity of such a small particle in a

short time, it will be a resultant velocity caused by both grav-

itational settling and Brownian motion.

The local deposition rate of particles by Brownian diffu-

sion onto a unit surface area, the deposition flux j (number of

deposited particles per unit time and surface area), is given by

j  – D N 
 vN 
 uN. (19)

If the flow is turbulent, the value of the deposition flux of

uncharged particles depends on the strength of the flow

field, the Brownian diffusion coefficient, and gravitational

sedimentation.

Particle Charging and Electrical Properties

When a charged particle having n p elementary charges is sus-

pended in an electrical field of strength E, the electrical force

F e exerted on the particle is n p eE, where e is the elemen-

tary charge unit ( e  1.6 10 ^19 C). Introducing F e into the

right hand side of the equation of particle motion in Table 3

and assuming that gravity and buoyant forces are negligible,

the steady state velocity due to electrical force is found by

equating drag and electrical forces, F d  F e. For the Stokes

drag force ( F d  3 pmv e D p / C c ), the terminal electrophoretic

velocity v e is given by

v e  n p eEC c /3pmD p. (20)

B e in Figure 3 is the electrical mobility which is defined

as the velocity of a charged particle in an electric field of

unit strength. Accordingly, the steady particle velocity in an

electric field E is given by Eb e. Since B e depends upon the

number of elementary charges that a particle carries, n p , as

seen in Eq. (3.7), n p is required to determine B e. n p is predict-

able with aerosol particles in most cases, where particles are

charged by diffusion of ions.

The charging of particles by gaseous ions depends on

the two physical mechanisms of diffusion and field charging

(Flagan and Seinfeld, 1988). Diffusion charging arises from

thermal collisions between particles and ions. Charging occurs

also when ions drift along electric field lines and impinge upon

the particle. This charging process is referred to as field charg-

ing. Diffusion charging is the predominant mechanism for

particles smaller than about 0.2 m m in diameter. In the size

range of 0.2–2 m m diameter, particles are charged by both dif-

fusion and field charging. Charging is also classified into bipo-

lar charging by bipolar ions and unipolar charging by unipolar

ions of either sign. The average number of charges on particles

by both field and diffusion charging are shown in Figure 4.

When the number concentration of bipolar ions is sufficiently

high with sufficient charging time, the particle charge attains

an equilibrium state where the positive and negative charges

in a unit volume are approximately equal. Figure 5 shows the

charge distribution of particles at the equilibrium state.

Reynolds number and the basic equation expressing the par-

ticle motion in a gravity field.

The terminal settling velocity under gravity for small

Reynolds number, v t , decreases with a decrease in particle

size, as expressed by Eq. (3.1) in Figure 3. The distortion

at the small size range of the solid line of v t is a result of

the slip coefficient, C c , which is size-dependent as shown in

Eq. (3.2). The slip coefficient C c increases with a decrease

in particle size suspended in a gaseous medium. It also

increases with a decrease in gas pressure p as shown in

Figure 3. The terminal settling velocities at other Reynolds

numbers are shown in Table 3.

t g in Figure 3 is the relaxation time and is given by

Eq. (3.6). It characterizes the time required for a particle to

change its velocity when the external forces change. When

a particle is projected into a stationary fluid with a velocity

v o , it will travel a finite distance before it stops. Such a dis-

tance called the stop-distance and is given by v 0 t g. Thus, t g

is a measure of the inertial motion of a particle in a fluid.

Motion of a Small Diffusive Particle

When a particle is small, Brownian motion occurs caused

by random variations in the incessant bombardment of mol-

ecules against the particle. As the result of Brownian motion,

aerosol particles appear to diffuse in a manner analogous to

the diffusion of gas molecules.

The Brownian diffusion coefficient of particles with

diameter D p is given by

D  C c kT /3 pmD p (15)

where k is the Boltzmann constant (1.38 10 ^16 erg/K) and

T the temperature [K]. The mean square displacement of a

particles ^ x^2 in a certain time interval t, and its absolute value

of the average displacement  x, by the Brownian motion, are

given as follows





xDt

xDt

(^22)
4

 ⁄p
(16)
The number concentration of small particles undergoing
Brownian diffusion in a flow with velocity u can be determined
by solving the following equation of convective diffusion,

 

N
t
⋅⋅uvNDN^2 N (17)
vFτgp∑ ⁄m (18)
where N is the particle number concentration, D the Brownian
diffusion coefficient, and v the particle velocity due to an
external force F acting on the particle.
The average absolute value of Brownian displacement
in one second,  x, is shown in Figure 3, which is obtained
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