Handbook of Solvents - George Wypych - ChemTech - Ventech!

364 Semyon Levitsky, Zinoviy Shulman
vapor presence, the pressure and/or temperature variations inside the oscillating bubble
cause evaporation-condensation processes that are accompanied by the heat exchange. In
polymeric solutions, transition from liquid to vapor phase and conversely is possible only
for a low-molecular solvent. The transport of the latter to the bubble-liquid interface from
the bulk is controlled by the diffusion rate. In general case the equilibrium vapor pressure at
the free surface of a polymeric solution is lower than that for a pure solvent. If a bubble contains
a vapor-gas mixture, then the vapor supply to the interface from the bubble interior is
controlled by diffusion rate in the vapor-gas phase. The concentration inhomogeneity
within the bubble must be accounted for if tDg >t0or PeDg >1
where:
2
tDg characteristic time of binary diffusion in vapor-gas phase, tDg= R0/Dg Dg diffusion coefficient
PeDg diffusion Peklet number for the vapor-gas phase, PeDg= tDg/t0 Fast motions of a bubble surface produce sound waves. Small (but non-zero) compressibility
of the liquid is responsible for a finite velocity of acoustic signals propagation
and leads to appearance of additional kind of the energy losses, called acoustic dissipation.
When the bubble oscillates in a sound field, the acoustic losses entail an additional phase
shift between the pressure in the incident wave and the interface motion. Since the bubbles
are much more compressible than the surrounding liquid, the monopole sound scattering
makes a major contribution to acoustic dissipation. The action of an incident wave on a bubble
may be considered as spherically-symmetric for sound wavelengths in the liquid lf>>R0. When the spherical bubble with radius R0 is at rest in the liquid at ambient pressure,
pf0, the internal pressure, pin, differs from pf0 by the value of capillary pressure, that is
p = p + 2σ/ R
[7.2.34]
in f 0 0
where:
σ surface tension coefficient
If the system temperature is below the boiling point at the given pressure, pf0, the thermodynamic
equilibrium of bubble in a liquid is possible only with a certain amount of inert
gas inside the bubble. The pressure in vapor-gas mixture follows the Dalton law, that suggests
that both the solvent vapor and the gas are perfect gases:
( )
p = p + p = ρ B + ρ B T = ρ B T , ρ = ρ + ρ [7.2.35]
in g v g g v v m m m m m g v
( )
B = 1− k0 B + k0 B , B = R / μ
m g v gv , g gv ,
where:
k0 equilibrium concentration of vapor inside the bubble
μ g,v molar masses of gas and solvent vapor
v index, referring to vapor
From [7.2.34], [7.2.35] follows the relation for k0: −1
−1
[ 1 v g{ ( 1 2σ) v0 1}
] v0 v0 f 0 σ σ ( v0
0)
k = + B B + / p − , p = p / p , = / p R [7.2.36]
0
The equilibrium temperature enters equation [7.2.36] via the dependence of the saturated
vapor pressure, p v0, from T 0. Figure 7.2.1 illustrates the relation [7.2.36] for air-vapor
bubbles in toluene. 25 The curves 1- 3 correspond to temperatures T 0 = 363, 378, 383.7K (the