1. Coagulation - definitions
when relative motion of particles is Brownian, process = thermal coagulation
when relative motion arises from external forces (eg gravity, electrical forces, aerodynamic effects) = kinematic coagulation
coagulation of solid particles = agglomeration

2. Collision frequency function
collision frequency - # collisions/time between particles
of size i and size j =
vi, vj are volumes of
particles of size i and j
b depends on the size of the colliding particles, and properties of
system such as temperature and pressure
Consider the change in number concentration of particles of size
k, where vk = vi + vj

3. Coagulation - discrete distributions
For a discrete size distribution, the rate of formation of
particles of size k by collision of particles of size i and j, is
given by: where the factor 1/2 is introduced
because each collision is counted
twice in summation
Rate of loss of particles of size k by collision with all other
particles is given by:
Change in number concentration of particles of size k given by:
theory of coagulation for discrete spectrum developed by
Smoluchowski (1917) change = formation - loss

4. Simple example

5. Collision frequency functions
for particles in
continuum regime:
(Stokes-Einstein
relationship valid)
for particles in
free molecular regime: (derived from kinetic theory of collisions
between hard spheres)
interpolation formulas between regimes given by
Fuchs (1964) The Mechanics of Aerosols

6. Collision frequency values:
where particle diameters, d1 and d2 are in microns

7. Coagulation: simple test case
Assume we have a monodisperse population initially, and
particle diameter is greater than gas mean free path
(continuum regime), so vi = vj
coagulation equation is then:
let
coagulation equation simplifies to:

8. Simple test case continued
solving for total number concentration as f(t)
using the boundary condition:
solving for time

9. Example problem:
Estimate the time for the concentration of a monodisperse aerosol
to fall to 10% of its original value for an initial size of dp =
1 micron and for the following initial concentrations:
103 and 108 # / cm3

10. Coagulation, continuous distributions
continuous nomenclature
n(v) = number of particles per unit volume of size v, a continuous distribution
collision rate:
where b is the collision frequency function described earlier
The rate of formation of particles of size v by collision of smaller
particles of size and is given by:
Here, loss is from collisions with all other particles,
so must integrate over entire size range

11. Continuous distributions continued
How to solve?

12. Recall: similarity transformation
The similarity transformation for the particle size distribution
is based on the assumption that the fraction of particles in a
given size range (ndv) is a function only of particle volume
normalized by average particle volume:
defining a new variable, and rearranging,

13. Self-preserving size distribution
For simplest case: no material added or lost from the system,
V is constant, but is decreasing as coagulation takes place.
If the form of is known, and if the size distribution
corresponding to any value of V and is known for
any one time, t, then the size distribution at any other time can be
determined. In other words, the shapes of the distributions
at different times are similar, and can be related using a scaling
factor. These distributions are said to be ‘self-preserving’. t1 t2 t3

14. Special cases - free molecular regime, coagulation of spheres
Brownian coagulation in free molecular regime: Lai, Friedlander, Pich, Hidy, J. Colloid Int. Sci. 39, 395 (1972).
allows estimation of change in total number concentration resulting from coagulation, if a self-preserving distribution is assumed
a is an integral function of y, and is found to be about 6.67
Note V =volume fraction,
vol aerosol /
total volume gas plus aerosol

15. Coagulation of hard spheres
When primary particles collide and stick, but do not coalesce, irregular structures are formed
agglomerate spherical equivalent
Recall - we can use a fractal dimension to characterize these structures
Above equation gives: the relationship between radius r (rgyration usually) of aerosol agglomerates, and the volume of primary particles in the agglomerate

16. Special cases - free molecular growth of fractal agglomerates
The change of agglomerate size distribution with time also can be described using a self-preserving size distribution scaling law.
Using a similarity transform, and extrapolating out to long times, average agglomerate radius grows as a power law function of time: (Matsoukas and Friedlander, J. Colloid Int. Sci. 146, 495 (1991)
and

17. Interesting result!
For Df = 3, growth rate is independent of primary particle size.
But, for Df < 3, then the exponent q is negative
So smaller primary particles result in larger agglomerates!

18. Summary of processes affecting suspended particles
Transport by diffusion and external fields
Formation by nucleation
Condensation and evaporation
Coagulation
Balances: