Environmental Engineering Reference
In-Depth Information
n
=
∑
ideal vapor. Therefore, the full partition function for the system of volume
V
consisting of
N
n
drops and
N
1
ideal vapor monomeric molecules can be written in the following form [31]:
⎛
⎞
N
q
N
∑
∑ ∏
n
n
id
Q V
(
)
=
Q
V
−
N
v
(19.55)
⎜
⎟
N N
,
n n
1
n
!
⎝
⎠
n
{ }
N
n
=
2
n
=
2
n
id
is the partition function of ideal vapor. The irst sum in Equation 19.55 is over all
distributions
N
n
such that
where
Q
N
1
,
N
n
∑
N
n
N
+
n
=
N
(19.56)
1
n
=
2
where
N
is the total number of molecules in the system
The equilibrium distribution is found in the usual manner by inding the maximum term in the sum
of Equation 19.55 subject to the conservation condition, Equation 19.56. The result is [31]
⎩
1
⎭
(19.57)
N
=
q
exp
−
(
P
v
−
n
μ
v
)
n
n
n
k
T
B
where
P
is the ambient ideal gas pressure. The substitution of Equation 19.54 into Equation 19.57
yields
⎧
⎨
⎩
rest
⎫
⎬
⎭
Φ
f
+
P
k T
v
−
n
μ
K
n
n
v
N
=
N
1
exp
−
(19.58)
n
N
1
B
where
f
rest
+ = is the Gibbs free energy for the drop at rest. Equation 19.58 differs from
the CNT expression for the drop size distribution [1] (Equation 19.29) by the factor Φ
K
/
N
1
. Thus,
the classical Equation 19.29 is to be modiied by the correction factor Φ
K
/
N
1
and, as the nucleation
rate is proportional to the number
N
crit
of critical nuclei, the same free energy correction factor will
appear in the classical expression for the nucleation rate.
rest
P
g
n
n
n
19.7 COMPARISON WITH THE KUSAKA'S NUMERICAL SIMULATION RESULTS
To distinguish the Kusaka factor Φ
K
calculated numerically [11] from the correction factor eval-
uated analytically we denote the quantity
Φ
R
K
/
by Φ (see Equation 19.49). Combining
Equations 19.35, 19.49 through 19.50, and 19.53 we propose the following expression for the cor-
rection factor Φ/
N
1
:
K
(
Q
rot
Q
)
rot
,
l
3 2
/
3
K
⎛
⎜
5
β
⎞
⎧
⎨
⎩
⎫
⎬
⎭
Φ Φ
Q
Q
1
2
64
R
ρ
k T
⎛
⎜
3
2
h
k T
ν
⎞
⎟
⎛
⎜
h
k T
ν
⎞
⎟
R
rot
B
max
max
5
=
=
×
π
exp
1
−
exp
−
(19.59)
⎟
K
sat
3 2
/
3
2
N
N
Sn
(
π
)
σ
15
h
1
1
rot l
,
1
B
B
where
S
=
S
is the supersaturation ratio
n
sa
1
is the saturated vapor number density
n
sat
n
1
1
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