Transcript Coagulation - definitions
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
Collision frequency function
collision frequency - # collisions/time between particles of size i and size j =
v i , v j
are volumes of
N ij
n i
,
n j
particles of size i and j depends on the size of the colliding particles, and properties of Consider the change in number concentration of particles of size k, where
v k
=
v i + v j
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: 2 1 N ij 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: i 1 N ik Change in number concentration of particles of size k given by:
dn k dt
1 2
i
j
k N ij
i
1
N ik
1 2
i
j
k
ij
(
v i, v j
)
n i n j
n k
i
1
ik
(
v i, v k
)
n i
theory of coagulation for discrete spectrum developed by Smoluchowski (1917) change = formation - loss
Simple example
Collision frequency functions
for particles in continuum regime: (Stokes-Einstein relationship valid)
ij
2
kT
3 1
v i
1/ 3 1
v
1/ 3
j
v
1/ 3
i
v
1/ 3
j
for particles in free molecular regime: (derived from kinetic theory of collisions ij 3 4 6 kT p 1 v i 1 v j v i v j interpolation formulas between regimes given by Fuchs (1964)
The Mechanics of Aerosols
2
Collision frequency values:
10 10 0.1
cm 3 /sec 1.0
d 1 /d 2 0.01
0.1
1.0
0.01
18 240 3200 14.4
48 6.8
where particle diameters, d 1 and d 2 are in microns
Coagulation: simple test case
Assume we have a monodisperse population initially, and particle diameter is greater than gas mean free path (continuum regime), so v i
ij
2 3
kT
1
v i
1/ 3 = v j 1
v
1/ 3
j
v
1/ 3
i
v
1/ 3
j
8
kT
3
K
coagulation equation is then: dn k dt K 2 n n i j Kn k i 1 let i 1 n i N n i coagulation equation simplifies to:
dN
dt
-
K
2
N
2
Simple test case continued
solving for total number concentration as f(t) N 1 N KN 2 t using the boundary condition: N N at t 0 solving for time t N N KN 2 1
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: 10 3 and 10 8 # / cm 3
Coagulation, continuous distributions
continuous nomenclature n(
v
) = number of particles per unit volume of size
v
, a continuous distribution collision rate: N
v
where is the collision frequency function described earlier The rate of formation of particles of size v by collision of smaller particles of size
v
and is given by:
formation
in range dv
l
oss in range dv 1 2 0 0
v
(
v
,
v
˜
v
)n(
v
(
v
, ˜ )n( ˜ )n( ˜ )d ˜ d
v
˜
v
d
v
Here, loss is from collisions with all other particles, so must integrate over entire size range
Continuous distributions continued
n
et rate of formation of particles of size v 1 2 0
v
(
v
,
v
˜
v
)n(
v
˜
v
d
v -
0 =
t v
(
v
, ˜ )n( ˜ )n(
v v
d
v Dividing
through by d
v
gives
c
hange in size distribution resulting from coagulation = n
t
1 2 0
v
(
v
,
v
˜
v
)n(
v
˜
v -
0 (
v
, ˜ )n( ˜ )n(
v v
How to solve?
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 (nd
v
) is a function
only
of particle volume normalized by average particle volume: here, average particle volume =
v
nd
v
N
v v
d
v v
=
V N where V is total aerosol volume defining a new variable,
v v
n N 2 V N
v
V and rearranging,
Self-preserving size distribution
For simplest case: no material added or lost from the system, V is constant, but N is decreasing as coagulation takes place.
If the form of is known, and if the size distribution corresponding to any value of V and N 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
’. t 1 t 2 t 3
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).
dN dt
a 2 4 3 1 / 6 6
kT particle
1 / 2
V
1 / 6
N
11 / 6 Note V =volume fraction, vol aerosol / total volume gas plus aerosol • • allows estimation of change in total number concentration resulting from coagulation, if a self preserving distribution is assumed a is an integral function of , and is found to be about 6.67
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 v v o r r 0 D f where v o 4 3 3 r is the volume of the primary particle 0 Above equation gives: the relationship between radius r (r gyration agglomerate usually) of aerosol agglomerates, and the volume of primary particles in the
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) r r 0 q z z 1 2 D f D f 1 2 dynamic exponent and q D f 3 D f 4 3
Interesting result!
• For D f = 3, growth rate is independent of primary particle size. • But, for D f negative < 3, then the exponent q is • So smaller primary particles result in larger agglomerates!
Summary of processes affecting suspended particles
• Transport by diffusion and external fields • Formation by nucleation • Condensation and evaporation • Coagulation Balances: