Environmental Engineering Reference
In-Depth Information
where
−
1
=
⎛
⎞
⎟
⎛
⎜
⎞
⎟
dP
dn
β
dP
dn
β
χ
(19.18)
⎜
μ
α
,
n n
crit
=
Thanks to Equations 19.6 and 19.14 we can write down for the last factor in Equation 19.17:
dn
dR
dn
dR
dR
dR
π
ρ
β
⎛
⎜
d
dR
δ
⎞
⎟
e
S
2
=
=
4
m
R
1
+
(19.19)
e
S
e
S
where ρ
β
=
mn
β
. The GTKB equation (19.7) can be rewritten as
−
1
⎛
⎜
⎞
⎟
1
d
dR
σ
σ
2
m
2
Γ
Δ
m
S
S
=
1
+
(19.20)
Γ
R
R
Δρ
R
ρ
S
S
S
S
Using Equations 19.19 and 19.20 and accounting that for the gas-liquid nucleation Δρ ≈ ρ
β
we have
2
−
1
−1
⎛
⎜
⎞
⎟
⎛
⎜
⎞
⎟
m
σ
(
R
)
1
2
Γ
Δρ
m
⎛
⎜
d
dR
δ
⎞
⎠
S
S
Y
=
2
1
+
χ
1
+
(19.21)
⎟
β
2
2
ρ
R
4
π
R
R
S
e
S
S
Then, with the help of the Gibbs-Tolman equation [33]
⎡
⎢
2
⎤
⎥
x
(19.22)
Γ
m
= ρδ 1
Δ
+ +
x
3
where
x
= δ/
R
S
, Equation 19.3 transforms into the sought formula for Zeldovich factor:
−
1 2
/
⎛
⎜
d
dR
δ
⎞
⎟
m
R
σ
R
k T
(
)
2
−
1 2
/
1 2
/
S
S
Z
=
[
1
+
x f x
( )]
χ
1
+
(19.23)
2
ρ
2
π
S
e
B
where
f
(
x
) = (1 + (2/3)
x
) (1 +
x
)
−2
. Substituting Equation 19.23 to Equation 19.1 and taking Equation
19.2 into account we have for the nucleation rate:
−
1 2
/
2
m
σ
(
R
)
1
⎛
⎜
d
dR
δ
⎞
⎟
⎛
⎜
W
⎞
⎟
2
2
S
S
−
1 2
/
1 2
/
crit
k T
I Kn
=
[
1
+
x f x
( )]
χ
1
+
exp
−
(19.24)
1
β
π
ρ
S
where
n
1
=
N
1
/
V
. Equation 19.24 includes the correction factor
K
which will be considered in the fol-
lowing sections. The only undetermined factor in Equation 19.24 is χ which is (see Equation 19.18)
the ratio of derivatives of pressure with respect to the number of molecules
n
for equilibrium and
non-equilibrium processes. It is natural to assume that this ratio is close to unity: χ ≈ 1. The values
x
= δ/
R
S
and
d
δ/
dR
S
in Equation 19.24 are related to σ
S
(
R
S
) by GTKB Equation 19.5 and can be cal-
culated if σ
S
(
R
S
) is known. Besides, for the large enough drops (which match the inequality
x
2
<< 1)
the quantities
x
2
f
(
x
) and
d
δ/
dR
S
can be neglected with respect to unity; that is, in this case Equation
19.24 is practically the same as classical Equation 19.9 (except for the factor
K
).
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