Collision and Coagulation of Particles
Important especially for the case of high concentration
Particle collision leads to a reduction in total number and an increase in the average size
Influence the determination of particle growth &morphology
Why does collision occur ?: difference in particle velocity (in vector)
(Brownian motion, Shear flow, turbulent motion,
Differential sedimentation, External force(electrically, acoustically))
Collision and Coagulation of Particles
Fast coalescence limit : can assume spherical particle (competition between collision and coalescence)Nij : number of collisions per unit time per unit volume between particles of volumes (vi, vj) : (unit #/cm3 sec)
: collision frequency function or coagulation coefficient : cm3/sec
( , )
ij i j i j
N v v n n
Rate of formation of particles of size v
k,
2
, ,
1 2
1 ( ) .
2
due to the indistinguishability of two equal sized particles.) half particle: red : half particle: blue
( ) 1 ( ) (
2 2 4 4 4 4
i j k ij
i j k
i j i j
i i i i i
i j i i j
v v v N
v v n n
n n n n n
v v n v v
For the case of i=j,
2
2 ,
2
2 2
) 1
4 8
( ) ( ) 1
8 8 8 8 8 8 8 8 16
1 1 1 1
( ), sin 2
4 8 16 2
1 as a result of indistinguishability of coagulating particles
i
i
i i i i i i i i
i i i
n i
n
i
n n
n n n n n n n n
v v n
n ce
n
Rate of loss of particles of v
k1
1
, ,
1
loss = 1 2
1 ( ) ( )
2
Particle dynamics equation for discrete size when coagulation alone is considered.
ik i
k
ij ij
i j k i
i j i j k i j i
i j k i
N
dn N N
dt
v v n n n v v n
Coagulation mechanisms
Brownian Coagulation
Laminar Shear
Turbulent Shear
Differential Sedimentation
Brownian coagulation
: important for dp << 1㎛0
1 continuum approach
: thermal motion of gas molecules and its associated random motion of small particles
1. equal-sized particles of radius at concentration : imagine o
g p
p
d m
r n
2
0 0
2
0
ne particle to be stationary
: point particle deposition on the fixed particle
2 ( , 0) ( , ) (2 , ) 0
2 2
Sol. ( , ) 1
2
p
p p
n n n
D n r n n t n n r t
t r r r
r r r
n r t n erfc
r Dt
The rate at which particles arrive at the surface r=2r
p2
0 2
0
16 8 1 2
initially very rapid collision, but
2 1,
collision rate approaches the steady state 8
p
p
p p
r r
p
p
n r
J r D r Dn
r Dt
r
Dt r D n
When rp1 and rp2 have Brownian motion simultaneously,
t x x t
x t
x t
x x
D
ij i j i j i j2
) 2 (
2 2
2
2 2
2
When there are more than one #1 particle, total collision rate between #1 and #2 particles per unit volume
1 1 12 20 10
,
, 1 2 1 2
1 2 1 2
1 1
3 3
1 2
1 1
3 3
1 2
1 2
1 2
4 ( ) #/cm3 sec
( )
( ) 4 ( )( )
2 ( )( )
2 1 1
3
3 3
p p
ij i j i j
i j p p
p p
p p
J r r D n n
N v v n n
v v D D r r
D D d d
kT v v
v v
kT kT
D D
d d
free-molecular regime, i.e., kinetic theory
dp g
3
32 2
( ) :mean number of particles per unit volume with center of mass velocity in the range between v and
( ) exp
2 2
: Maxwell-Boltzman f v d v
v dv
m mv
f v n
kT kT
n distribution
3
3
3 2
2 2
0 0 0
12 3
0
1 ( )
in spherical coordinate sin
1 sin ( )
4 8
( ) : mean speed
x y z
v vf v d v n
d v dv dv dv
d v v dv d d
v v dv d d f v v
n
f v v dv kT
n m
Collision rate per unit area
area A
→ Z
#/m sec : number of collisions per unit area per unit time2
i) crude calculation
number of particle having Z-direction velocity per unit volume
6
particle wi
n
th moves in , which makes volume total number of particles which strike
per unit area per unit time 1
6
v vdt dt vdtA
nv nv
3
ii) Exact calculation : consider the distribution of particle velocity consider particles with velocity v to v+dv its
direction to and to
( ) cos :number of particles of this t
d d
f v d v vdt dA
3 0
ype that strike the area dA in time dt total number of particles that strike a unit area of the wall per unit time
( ) cos
0 0 2 0
2 2
: particles leaving from f v v d v v
3 2
3 2 2 3
0 0 0 0 0
3
0
the wall sin
( ) sin cos ( )
4 1
( ) Effusion flux
4 d v v dv d d
f v v dv d d f v v dv
v f v v dv nv
n
1
2 2 12
12 1 2
2
1 2 2 12
2
12 1 2 2 12 1
2
1 2 12
Fix particle
:mean speed
1 4
collision rate per single particle of 1
total collision rate between 1 and 2 particles per unit volume 1
4 4
p
p p
p p
p p
d
v v v
d d n v
N d d n v n
d d v
12 12
2 2
12 1 2 1 3 3
2 1 2
1 1 2 2
2
1 2 3 3
1 2
3 3
1 1 2 2
1 1
3 3
3 3
1 2
1 2
1 1
16 2 2 1 1
3 3
4 3 1 1
3 1 1
1 1
6 6
6 6
3 6 1 1
4
p p
p
p p
p p p
p p
p p
i j
p i j
v v v kT
d d
kT d d
d d
v d v d
v v
d d
kT v v
v v
2
Eq. (7-17)
Transition regime? Fuchs formula
1
1 2 1 2
12 1 2 1 2
1 2 12 12 1 2
2 3
2
TABLE 10.1 Fuchs Form of the Brownian Coagulation Coefficient K12
8( )
2 ( )( )
2 ( )
5 4 6 18
Phillips(1
3 5 (8 )
p p p p
p p
p p p p
i i i
i
pi i i
D D D D
K D D D D
D D g c D D
Kn Kn kn
D kT
D Kn Kn
3
1 3
2 2 2 2 2 2
12 1 2 1
12
1
2 2 12
12 1 2
975) 1(3 )
8 8 2
pi i pi i pi
pi i
i air
i i i
i pi
g g g g D l D l D
D l
D kT
l c Kn
m D
c
c c c
Fig.13 Brownian coagulation coefficient K(Dpi, Dpj) for particles of density p = 1 g cm-3 in air at 298K, 1atm
1 2
12
12
continuum regime
8 ; independent of particle size 3
free molecular 4 6
The maximum occurs near 0.02 ( 1 / ) The reason is that large particle has lo
p p
p p
p
d d
kT
kT d
m g cc
w diffusivity and small particle has small cross-section area
The maximum near 0.02m
2 1
1 2
2
2 2
2 2
1 2
For given , if becomes larger, target area increases proportionally to but, diffusivity decreases with 1
2 1
lim 3
for free molecular regime, lim 3
p p
p p
p
p p
p p
p p
d d
d d
d d
kT d
d d
d d
12 2
2 32 1 p
p p
kT d
d
Coagulation coefficient increases more rapidly with dp2 for free molecular regime than for
continuum regime
