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Encyclopedia of Physical Science and Technology En001c-14 May 7, 2001 18:25

Aerosols 283

and R g /L g 1, where λ g is the mean free path of the
dN = N g ( , ˜ n 2 ··· ˜ n k , r, t) d d ˜ n 2 ··· d ˜ n k
gas and L g a typical length scale of the system, such as a
Here, ˜ n i , the number of moles of a given species, has been spherical collector diameter or a pipe diameter. The theory
˜
eliminated from the function g by the relation, = i ˜ n i i , can be extended to incorporate electrical effects, as well
where ˜ is the partial molar volume of species i and r is as a coagulation or sticking capacity and gas-condensed
i
a position vector. Since the integral of dN over all ˜ and phase interactions.
˜ n i is N, Virtually all of the mechanical theory of particles
emerges from a simplification called the single-particle

··· g( , ˜ n 2 ··· ˜ n k , r, t) d d ˜ n 2 ··· d ˜ n k = 1 regime. In this situation, particles are assumed to interact
only “instantaneously” in collision; otherwise, they can
˜ n k
be assumed to behave as a body moving in a medium of
Furthermore,thesizedistributionfunctioncanberetrieved
infinite extent.
by integration over all chemical species.
In general, the exchange of momentum between a gas
and a particle involves the interaction of heat and mass

n( , rt) = N ··· g( , ˜ n 2 ··· (˜ n k , r, t) d ˜ n 2 ··· d ˜ n k transferprocessestotheparticle.Thus,theforcesactingon
˜ n 2 ˜ n k
a particle in a multicomponent gas containing molecular
The generalized distribution functions offer a useful gradients (nonuniform) may be linked with their gradients
means of organizing the theory of aerosol characteriza- as well as velocity gradients in the suspending medium.
tion for chemically different species. To date, however, The assumption that Kn approaches zero greatly simpli-
the data have not been sufficiently comprehensive to war- fies the calculation of particle motion in a nonuniform
rant application of such formalism. gas. Under such circumstances, momentum transfer, re-
sulting in particle motion, is influenced only by aerody-
namic forces associated with surface friction and pressure
III. KINETIC THEORY OF AEROSOLS gradients. In such circumstances, particle motion can be
estimated to a good approximation by the classical the-
Historically a large segment of work in aerosol science ory of a viscous fluid medium where Kn is zero. Heat
has focused on the motion of particles in fluid medium and and mass transfer can be considered separately in terms
on the associated heat and mass transfer to that particle. of convective diffusion processes in a low Reynolds num-
Recent theory has recognized that significant differences ber regime (Re ≤ 1). In cases where noncontinuum effects
exist in momentum, heat, and mass transfer depending must be considered (Kn > 0), the coupling between par-
on the continuum nature of the suspending medium. This ticle motion and thermal or molecular gradients as in gas
is normally characterized by the ratio of the mean free nonuniformities becomes important, and socalled phoretic
path of the gas and the particle radius. This ratio is some- forces play a role in particle motion, but heat and mass
times called the Knudsen number (Kn). For very small transfer again can be treated somewhat independently.
Kn, particles behave as if they are suspended in a contin- Phoretic forces are associated with temperature, and gas
uum medium. For very large Kn, the suspending medium component concentration gradients, or electromagnetic
is highly rarified and the particles respond to individual forces.
collisions of the suspending gas molecules. It is common practice to treat particle motion as the
basic dynamic scale for transport processes. This is readily
illustrated for particles in steady rectilinear motion.
Particle Mechanics: The Gas Kinetic Model
Idealization of particle behavior in a gas medium involves
a straightforward application of fluid dynamics.
Mechanical constraints on aerosol particle dynamics A. Motion of Particles
can be defined by certain basic parameters. Model parti- 1. Stokes’ Law and Momentum Transfer
cles are treated as smooth, inert, rigid spheres in near ther-
modynamic equilibrium with their surroundings. The par- When a spherical particle exists in a stagnant, suspending
ticle concentration is very much less than the gas molecule gas, its velocity can be predicted from viscous fluid the-
concentration. The idealization requires that the ratio of ory for the transfer of momentum to the particle. Perhaps
the size (radius) of gas molecules (R g ) to that of particles no other result has had such wide application to aerosol
i, R g /R i , be less than 1 and the mass ratio, m g /m i 1. mechanics as Stokes’ (1851) theory for the motion of a
ApplicationofBoltzmann’sdynamicequationsforaerosol solid particle in a stagnant medium. The model estimates
behavior requires further that the length ratios R g /λ g 1 that the drag force acting on the sphere is