THETURBULENT COAGULATION OF COLLOIDS
April, 1956
463
TURBULENT COAGULATION OF COLLOIDS BY H. L. FRISCH Department of Chemistry, University of Southern California, Los Angeles 7 , Calif. Recebsd September 29, 1066
The theory of both rapid and slow turbulent coagulation of a colloidal suspension in a homogeneously and isotropically turbulent medium is developed in terms of the principal physical parameters characterizing the turbulence. The coagulation time, 9, aa expected, for large 8 satisfies a relation similar to one derived by Smoluchowski for coagulation aa a result of molecular Brownian motion, except that the molecular diffusion coefficient is replaced by an appropriate edd dif€usivity constant. For more rapid coagulations this similarity breaks down and the exact form of 9 is shown to dependTstrongly on the form of the Lagrangian correlation coefficient. The effect of boundary effects and repulsive forces due to electrostatic charges on the particles is considered.
I. Introduction-Turbulent Diffusion Consider a swarm of particles whose velocities form a field of homogeneous and isotropic turbulence in a fluid macroscopically a t rest.l We shall assume that there is no correlation between the molecular agitation and turbulent fluctuations in the positions of the suspended particles. T h u s we can write for the variances in the position of the particles yzht.l y’tot =
y’turb
+
(1)
y’mol
It has been shown‘ that if c(z, y, z; t ) is the mean concentration of particles located in a volume
element dv centered about
2, y,
z a t time t , then
2 = n*(t)vZc at where n* is defined t o be the factor of turbulent diffusion2 Idyl
-
n*(t) = 2 dt
(3)
Introducing the Lagrangian correlation coefficient &(h) of the field of turbulence we can write instead of eq. 3 n*(t) =
62
E
Rh(a)da
(44
where 3 is the variance of turbulent velocities. If we wish to take into account also molecular agitation for which
-
y4moi= 2Dt
D the diffusion (molecular) caefficient, eq. 4a becomes’ n*(t) = 62
E
Rh(cu)da
+D
(4b)
The purpose of this paper will be to describe the kinetics of the mutual coagulation of the suspended particles in terms of the parameters characterizing the turbulent motion of the particles themselves. The physical systems to which these results are to apply can be either hydrosols or aerosols being subjected t o extremely turbulent stirring without net macroscopic convection of the suspending medium. The viscosity of the suspending medium is taken (1) F. N. Frankiel, Advances in A p p l . Mech.. 111, GI (19.53). (2) F. N. Frankiel, Compf. rend., 994, 98 (1947).
to be sufficiently large so that in good approximation the motion of the suspended particles characterizes the motion of the medium itself. II. Rapid Turbulent Coagulation As a preliminary step to further consideration, we verify that, in view of the usual assumption’J of a Gaussian distribution for the probability -c
density *(r, t ) for the particles a t a time t, the probability of a relative displacement of twn particles, initially, together a t t = 0 which lies + + + between T and r dr is
+
Equation 5 follows directly from the random nature of the turbulence and is a direct analogy to a result in Brownian motion due to Smoluch~wski,~ whence
-
Y?,a =
2 +2
(6)
Combining eqs. 3 and 6, we find -
nt(t) = 2 1 dl = n:(t)
+ nf(t)
(7)
which is the basic result we require in the theory of coagulation. Following Smoluchowski4we suppose that (fast) coagulation results from the coalescence of particles collidingwith one another. By virtue of eq. 7 we can ca81culatethe flux of particles colliding with a given one of the swarm by holding that one fixed a t the origin of our coordinate system, where it is surrounded by an effective “sphere of influence” for coagulation of radius R I , ~ .Assuming further that our particles are distributed uniformly a t time t = 0 in a medium of infinite extent, we find the concentration of particles about our central one by solving, cf. eq. 2
c = co a constant a t t = 0 for r c = 0 at r = R1,*for 8 > 0.
>R
(3) G . K. Batchelor, “Theory of Homogeneous Turbulence.” Cambridge University Press, Cambridge, 1953. (4) M. v. Smoluchowski, 2. phyeik. Chem., 93, 129 (1917); nee also S. Chandrasekhar, Reu. Mod. Phys., 15, 1 (1943).
H. L. FRISCH
464
Vol. 60
to unity we find
We find
r = 4aR E n * ( a ) d a
= 2aRG(t) while more generally r
since bc
b2C 2 bc br2 + F G
by virtue of eq. 7. From eq. 8 the rate at which particles arrive a t the surface r = R I ,is~given by
An important physical parameter first introduced by Smoluchowski to describe a coagulation process is the coagulation time 9, where, cf. eq. 14 46) =
I/PCll
The corresponding quantity in turbulent coagulation depends markedly on the form of the Lagrangian correlation coefficient and satisfies by virtue of eq. 16 and 14 More generally, as a result of the coalescence of single, double, etc., particles, we can write for the specific rate of formation of a particle of size i k
+
Vi,kciCk dt
Two limiting relations can be derived, the first for very short times and the second for long times, respectively, without detailed knowledge of the turbulence field. For times long in comparison with the Lagrangian scale of turbulence Lh, where
where ci, Ck are the concentrations of particles of size i and k , respectively, and nCdt) = nP(t)
f $@Lk
+ nk*(t)
Lh
we can write
=
2
(lob)
& ( a ) da
-
2T2Lht or = [2v2La+2D]t = 2Kt
Y'turb.
= nitk
lom
-
Following Smoluchowski4 we can write for the variation of k-fold particles with time
where K is the constant eddy diffusivity. Neglecting the term O(S1/~)we have from eq. 17a
In order to integrate eq. 11we shall make the rather questionable assumption4 that there exists an average in some sense over i and j of Vi+j such that
This result goes over smoothly into the value in the absence of turbulence 9 = [4?rDRc0)--l. For times which are small in comparison to L h as well as the Lagrangian microscale of turbulence' Ah = [-'/zd2Rh(0)/dh2]-1/t one finds
d = (4?rKRCo)-', d
Vi+ j ( t ) = 2V(t) (i,j = 1, 2, . . .).
Solving we find4 Ck(t)
k = 1,2, and
...
= Co[(coT)k-'/(l
+
COT)k+11
(13)
>>Lh
(17b)
y'turb. = 0' t' or y2 = 3 t 2 + 2Dt
This result implies that for very short times t < D / F molecular agitation produced by Brownian motion is more important than the effects of turbulence. We find on substituting the above in eq. 17a that
m
C(t)
=
Ck(t) =
Co/(l f
COT)
(14)
Q =1
with co the number concentration of single particles present a t t = 0 T =:
V ( a )d a
-
=
(2?rR~o)-',a
>
where y is a length of the order of a mixing length which, for a small condensation coefficient a! between particles, varies inversely with a. I n the case of turbulence, y is itself a function of the time which we will for the sake of mathematical simplicity treat as roughly constant. Letting = yR/(R 7 ) we find that the appropriate solution of the diffusion equation is no longer eq. 8 but
+
465
whence it follows that the coagulation time 9 is given by (cf. eq. 14)
More generally 8 must be found as a solution of 7(8) =
co-1
(22b)
where r as a function of g(8)is given by eq. 21a and y”(t9) as a function of t9 by eq. 19.
IV. Slow Turbulent Coagulation We have restricted our discussion so far to “rapid” coagulation in the sense that any electrostatic repulsive forces between the “spheres of interaction” of two particles are sufficiently weak to be neglected. When this is not the case Fuchs6 has shown how to modify the Smoluchowski coagulation theory for the presence of these forces as long as these are derivable from a potential # ( r ) , i.e., the forces are conservative. For lyophobic colloids the form of #(r) is known and the effect of this correction is tabulated.’ When these forces are appreciable then the rate of coagulation is materially slowed down so that one refers to the process as one of “slow” coagulation. Extending these ideas to the case of turbulent coagulation there should be an analog of a slow coagulation process resulting from the mutual repulsion of the particles. In the case that 9, >> L h and if an analog to a friction coefficient f exists for the particle in turbulent flow and inertial effects can be neglected, then I?* =
6
2
(23)
Here Q is the coagulation time in the absence of repulsive forces and x is obtained by solving an appropriate Smoluchowski equation4 with the ordinary molecular diffusion coefficient replaced by the eddy diffusion coefficient K , since for large t n* +K
Using this solution we find that the concentration of k-fold particles as well as the total number of particles is still given by eq. 13 and 14, respectively, since their derivation is unchanged except for the definition of ~ ( t ) . Assuming as before that a suitable average radius R for the coagulating colloid can be defined, then we have instead of eq. 15
I n particular when [y2(1) 1/E2 > 1 we find that ~ ( tis) given asymptotically as
Thus for large times and R
