SELF-PRESERVING SIZE DISTRIBUTIONS
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This equation is similar in form to the one presented by Friedlander,’
who used the radius as the independent variable.
Another relation between N, and n(v) may be obtained by integrating Eq. 4 twice over all particle sizes to give twice the total num-
ber of collisions. Since each collision results in the loss of a particle,
the number of collisions per second is the rate of change of N,:
qgN. _ ef t n(v)n(¥)(v% + 74) (45 + 3) dv dv
(6)
Smoluchowski’s decay law for uniform-size particles can be obtained
from Eq. 6 by noting that, for v = ¥, (v*+ 7*)[(1/v%) + (1/¥%)] = 4, and
the equation reduces to
2
* = -*
af t n(v)n(v) dv ay = - N?
(7)
SELF-PRESERVING SIZE DISTRIBUTION
In principle,
the course of a coagulation process could be pre-
dicted theoretically by solving Eq. 5 by using a given initial distribution function n(v,0), Equation 5 is a nonlinear partial integro-differential equation, and an analytic solution is not known. However, it will be
shown in this paper that it is possible to reduce Eq. 5 to a total differential equation by means of a Similarity transformation. This is a
mathematical device by which a particular grouping of the independent
variables is chosen as a new independent variable. The transformation
makes it possible to express the differential equation and the boundary
conditions solely in terms of the new independent variable, Simultaneously it helps in the correlation of experimental data
and
gives
physical insight into the problem.
The basic assumption for this theory is that the distribution function, n(v,t), can be expressed as
nv,t) = ett) + (4)
(8)
where g(t) and v* are functions of time. It is assumed that the size
distribution function, 7%, does not change with time. Friedlander’
pointed out that when an equation of this form was substituted into the
coagulation—sedimentation equation, a total integro-differential equation was obtained. This showed that the assumed form represented a
particular solution of the equation but perhaps not the only one.