Environmental Engineering Reference
In-Depth Information
Table 14. Several necessary parameters used in Eqs. (5.15) and (5.16)
h
(nm)
E b
(kJ/g-atom)
T b
(K)
S b
(J/g-atom K)
γ lv0
(mJ/m 2 )
0.372
97.7
1156
Na
84.5
0.286
293.4
2792
Al
105
H 2 O
0.096
75
13.6
373
36.5
Ref
42,213
214
42,54
42,54
As shown in figures 11~13, γ lv ( D ) decreases with size, following the trend of γ sv ( D ) and
E ( D ) [97,162]. This is because γ lv ( D ) as an energetic difference between surface molecules
and interior molecules of droplets decreases as energetic state of the interior molecules
increases more quickly than that of the surface molecules. Note that, prediction of Eq. (5.15)
provides the same or better accuracy of Eq. (5.13).
Although δ is assumed to be a constant as required by the derivation of Eqs. (4.9) and
(5.13), several applications of statistical thermodynamics have indicated that δ depends
strongly on D [209,215-216]. Since the results of these treatments are based on rather
complex numerical calculations, it would be difficult to express δ( D ) analytically.
Fortunately, Eq. (5.15) can be used to satisfy this requirement.
90
60
Eq. (5.15): Current model
Eq. (5.13): Tolman model
Simulated [Ref. 214]
30
H 2 O
0
0
2
4
6
8
D (nm)
Figure 13. D dependence of γ lv ( D )/ γ lv0 with δ = h for water [214].
Substituting Eq. (5.15) into Eq. (5.13) rather than Eq. (4.9) because the latter is an
approximation of Eq. (5.13) and will lead to error when D /δ ≤ 10 [211],
D
2
S
1
1
.
(5.16)
δ
(
D
)
=
exp
b
1
1
4
3
R
2
D
/
h
1
2
D
/
h
1
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