SELF-PRESERVING SIZE DISTRIBUTIONS
257
The expression for (1/N%)(dN.,/dt) can be obtained by substituting
Eq. 11 into Eq. 6, which gives, after some simplification,
”
YF daNn,_ kT
3n e+e f n°
Nodt
“
—%
wh
pe
~
(15)
dyn) anf Aj 1h) a
This expression substituted into Eq. 14 gives the final form
3 + f
0
1
,
7% v4) «| b din) +n an
78 bin) ay fP
7
~
me,
[ow
~
1
1
+5 f VG) vin — 7) [94 + — 99] li tao
|a
- van) f
Pw
wofl
ne
11.
v4) [08 = #5]25 +35a = 0
(16)
The partial integro-differential Eq. 5 has been transformed into an
ordinary integro-differential equation by the substitution of Eq. 11.
With the assumption that a solution exists, the new equation must be
solved subject to the integral conditions shown in Eqs. 13a and 13b.
The self-preserving form, Eq. 11, represents a particular solution to
the kinetic equation for coagulation by Brownian motion, It can be
shown that Eq. 11 is not the most general solution to the coagulation
equation. Both computer calculations and experiments indicate that the
self-preserving form is an asymptotic limit approached by the actual
solution,
One can gain some understanding: of the coagulation of a hetero-
disperse hydrosol without solving Eq. 16. For a monodisperse hydrosol
whose coagulation is described by Smoluchowski’s model, the coagulation rate is given by
aN,
— 4kT N?
dt 3
By rearrangement of Eq. 15, the rate of coagulation of a dispersion
with a self-preserving size distribution can be expressed as
dN.
dt
kT
eo
"ey
~
=- 3
Jove
f
n7*
49)
an
f
7
by)
a
Ne,
3
0
0
(17)
and, because the integrals are independent of time, the quantity
Ay = [242 [a7 dln) dn [> #8 ds) oF]
(18)