Environmental Engineering Reference
In-Depth Information
cluster radius
r
0
and the mean free path
for atoms in a gas, and the mobility and
diffusion coefficient of clusters in gas for these cases are given by (2.19), (4.113),
and (4.114).
The relaxation character of a cluster in a gas is connected to the large cluster
mass compared with the mass of an atom, and hence equilibrium of clusters with
a gas results from many collisions with atoms of a gas. In particular, for a rareness
gas (2.19) represents the motion equation for a cluster in the form
λ
d
dt
D
ν
w
,
where
w
is the cluster velocity with respect to the gas in which it is located and
is
the rate of relaxation. If the relaxation rate for gas atoms is determined by the rate
of collisions with other atoms, that is, atom relaxation is determined by a single
collision, many collisions are required for a cluster to change its velocity because of
the large cluster mass. Hence, the rate of cluster relaxation decreases with increas-
ing cluster size. In particular, within the framework of the liquid cluster model, for
the relaxation rate of a large cluster we have [60, 61]
ν
8
p
2
8
N
a
r
2
W
p
2
mT
3
nm
a
π
π
mT
N
a
r
0
D
n
1/3
,
ν
D
(6.1)
3
m
a
where
m
is the mass of a gas atom,
m
a
is the mass of a cluster atom, and
n
is the
number of cluster atoms.
Long relaxation of the cluster velocity if the cluster velocity differs from the mean
gasvelocityisdeterminedbythelargeclustermass.Thistakesplaceingasflows
with clusters where the gas velocity varies. In particular, we consider the spread
case when clusters are in a gas flow in a tube of variable cross section as shown
in Figure 6.1. Then the gas flow leaves the tube through an orifice and acquires a
velocity
c
s
of the order of the sound velocity near the orifice. Clusters in this flow
do not influence the character of this laminar flow, but in the case
0
c
s
ν
1
the drift velocity of clusters near the orifice is low compared with the flow velocity
c
s
. In particular, in the case of a cylindrical tube and a conic exit, as shown in
Figure 6.1
The geometry of a cylindrical tube with a conic exit through which a gas with clusters
flows.
R
0
is the radius of the cylindrical tube,
0
is the orifice radius, and
α
is the conic angle.
