Page 59 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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

282 Aerosols

sizes can only be multiples of a singular species. As an ex- where V is the volume fraction of particles in cubic cen-
ample, consider the coagulation of a cloud of particles ini- timeters of material from cubic centimeters of gas. If the
tially of a unit size. Then, after a time, all subsequent par- particle density is uniform, the average particle volume is
ticles will be kth aggregates of the single particle, where
V 4 M 3
k = 1, 2, 3,... represents the number of unit particles per ¯ = = π
N 3 M 0
aggregate. Physically, the discrete size distribution is ap-
pealing since it describes well the nature of the particu- The volume mean radius is 3M 1 /4π M 3 .
late cloud. The second function, continuous distribution, In general, particle distributions are broad in size–
is usually a more useful concept in practice. This func- concentration range so that they often are displayed on
tion is defined in terms of the differential dN, equal to the a logarithmic scale. For example, data are frequently
number of particles per unit volume of gas at a given point reported as log n R versus log R. Another display is
in space (r) at time t in the particle volume range, and dV/d log R versus log R. The area under the distribution
+ d . The distribution function then is curve plotted in this way is proportional to the mass con-
centration of (constant density) particles over a given size
dN = n( , r, t) d range, independent of size.
Although this form accounts for the distribution of par- The shape of the size distribution function for aerosol
ticles of arbitrary shape, the theory is well developed for particles is often broad enough that distinct parts of the
spheres. In this case, one can also define the distribution function make dominant contributions to various mo-
function in terms of the particle radius (or diameter), ments. This concept is useful for certain kinds of practi-
cal approximations. In the case of atomospheric aerosols;
dN = n R (R, r, t) dR
the number distribution is heavily influenced by the ra-
where dN is the concentration of particles in size range dius range of 0.005–0.1 µm, but the surface area and vol-
R and R + dR, n R is the distribution function in terms of ume fraction, respectively, are dominated by the range
radius, and t is time. The radius may be geometric, or it 0.1–1.0 µm and larger. The shape of the size distribution
may be used as an aerodynamic or other physical equiv- is often fit to a logarithmic-normal form. Other common
alent. An aerodynamic radius is defined in terms of geo- forms are exponential or power law decrease with increas-
metric size and particle density, which govern the motion ing size.
of the particle. Other physical parameters are the optical The cumulative number distribution curve is another
equivalent radius, which depends on the light-scattering useful means of displaying particle data. This function is
cross section of the particle. defined as:
The volume and radius distribution functions are not R
equal, but can be related by the equation: N(R,r, t) = n R (R,r, t) dR
0
2
n R = 2π R n It corresponds to the number of particles less than or equal
to the radius R. Since n R = dN(R)/dR the distribution
The moments of the size distribution function are useful
function can be calculated in principle by differentiating
parameters. These have the form:
the cumulative function.
∞

a
M(r, t) = n R R dR
0 C. Chemical Properties
The zeroth moment (a = 0),
The chemical properties of particles are assumed to corre-
∞ spond to thermodynamic relationships for pure and mul-

M 0 = n R dR = N
0 ticomponent materials. Surface properties may be influ-
enced by microscopic distortions or by molecular layers.
represents the total number concentration of particles at a
Chemical composition as a function of size is a crucial
given point and time. The first moment normalized by the
concept, as noted above. Formally the chemical composi-
zeroth moment gives a number average particle radius:
tion can be written in terms of a generalized distribution
∞ ∞

¯ function. For this case, dN is now the number of particles
M 1 /M 0 = R = n R Rd R n R dR
0 0 per unit volume of gas containing molar quantities of each
chemical species in the range between ˜ n i and ˜ n i + d ˜ n i ,
The third moment is proportional to the total volume con-
with i = 1, 2,..., k, where k is the total number of chem-
centration of particles, or
ical species. Assume that the chemical composition is dis-
4 4 ∞ 3
V = π M 3 = π n R R dR tributed continuously in each size range. The full size–
3 3 0 composition probability density function is