254
FRIEDLANDER
variable. Because of certain simplifications in the equations and the
fact
that the experimental distributions were in terms of particle
volume, the-volume will be used for this presentation, The continuous
distribution function, n(v,t)* (where v is the particle volume in cubic
centimeters and t is the time in seconds), is defined such that n(v,t) dv
is the particle concentration in the volume range from v to v + dv, It
follows that the total number of particles per cubic centimeter, N,,,
is defined by
N= {
“0
nv) dv
(1)
and the number of particles per cubic centimeter larger than a given
volume, N,, is defined by
Ny = [” n(v) dv
(2)
The total volume fraction of particles is given by
p= f vn(v) dv
(3)
where > is the phase volume ratio (volume of dispersed phase per unit
volume of continuous phase), This fraction is constant in an experiment
because no second-phase material is lost or gained. It is to be under-
stood that n(v), N., and N, are all time-dependent functions, but this
will not always be shown explicitly henceforth. The Brownian-motion
collision frequency between particles of volumesv and V is
2kT
= 3 (3, + =) (v3 + F4)n(v)n(¥) dv dF
va
b
(4)
where k = Boltzmann’s constant
T = temperature
uw = fluid viscosity
With the use of this expression for the collision frequency, an
equation for the rate of change of the distribution function with time
can be written as follows:
dn(jv) kT
dt
3
J,
(*
_2kT
341 n(v) /~
*Particles/cm®,
~.|
1
1
~
why
ane
n(¥)n(v — ¥) ae aa [V7 + (v — ¥)7] av
(2,4)
ages 7+ ¥ vy (a
,} dvae
n(v)(v
(5)