Environmental Engineering Reference
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This gives
Z
Z g z gw z @'
m a g z g
T
1
N
1
N
( g )
h g g
iD
g g
'
( v ) d v D k
d g D w
@
g z
m a
3 T h
g 3
D w
i
,
where k is the unit vector along the direction of the cluster velocity. From this we
obtain for the resistive force in the kinetic regime when a cluster moves in a gas
with velocity w [93]
8 p 2
m a
3 T
π
m a T
3
Nr 0 .
F D w
λ h v
iD w
(4.112)
3
Accounting for the connection between the cluster velocity w and the electric
field strength E
w D
K E ,
and taking the resistive force to be F D
e E foraclusterofcharge e moving in an
electric field of strength E , we obtain on the basis of (4.112) in the kinetic regime
of cluster drift [94]
3 e
K 0
n 2/3 , K 0
3 e
K
D
8 p 2
m a TNr 0 D
D
8 p 2
.
(4.113)
m a TNr 2 W
π
π
This expression coincides with the first Chapman-Enskog approximation (4.21).
From this on the basis of the Einstein relation (4.38) we obtain for the cluster
diffusion coefficient D in the kinetic regime of cluster drift [94]
3 p T
8 p 2
3 p T
8 p 2
D 0
n 2/3
D
D
m a Nr 0 D
, D 0
D
,
λ
r ,
(4.114)
m a Nr 2 W
π
π
which also corresponds to the first Chapman-Enskog approximation (4.53). Ta-
ble 4.16 gives the values of the reduced diffusion coefficients of metal clusters in
some gases in accordance with (4.114) in the kinetic regime of cluster drift.
Let us compose the ratio between the cluster diffusion coefficient D kin (4.114) for
the kinetic regime of cluster movement in a gas and the cluster diffusion coefficient
D dif for the diffusion regime of cluster movement according to (4.110) using (4.59)
for the viscosity coefficient. We have
D dif
D kin D
π
64 p 2
45
r 0 D
D r 0
1.56Kn ,
Kn
,
(4.115)
where Kn is the Knudsen number. For the cluster diffusion coefficient this expres-
sion combines both the kinetic and diffusion regimes of cluster drift:
D
D
D dif (1
C
1.56Kn) .
(4.116)
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