b
liq
dr
gv
Р
=
,
(22)
where
liq
is the density of the liquid phase, kg/m
3
; g is the free fall acceleration,
m/s
2
;
b
v
is bubble volume, m
3
.
The medium resistance
.
.res
m
Р
acts simultaneously with the beginning of bubble
displacement. Obviously, both the driving factor and the resistance factor depend on the
bubble volume, which determines the dynamics of the transition process, the completion
of which corresponds to the equality of
dr
Р
and
.
.res
m
Р
. The interaction between the bubble
mass
b
m
of the liquid phase corresponds to the equation:
.
.res
m
dr
b
Р
Р
у
m
−
=
,
(23)
where
у
is the acceleration of the bubble mass, m/s
2
.
Substitution of
dr
Р
and
.
.res
m
Р
values leads to:
2
2
у
f
gv
у
m
b
b
liq
b
−
=
,
(24)
where ξ is the coefficient depending on the properties of the medium;
b
f
is the bubble
projection area perpendicular to the direction of its movement;
у
is the bubble ascension
speed.
Condition (24) is a nonlinear differential equation of the second order, the initial
conditions of which are:
()
()
()
0
;
0
;
0
=
=
=
in
in
in
у
у
f
.
(25)
The autonomous processes of dispersed gas phase formation are accompanied by a
rather wide range of gas bubble sizes in the range from 0.5 to 8-15 mm in diameter,
depending on the level of turbulence in local zones, which affects the divergence of the
results of the equation (24), although it is generalized by the graphical interpretation
presented in Fig. 1.
Further ascension of gas bubbles is
accompanied by a decrease in hydrostatic
pressures, which leads to an increase in
volumes, sizes and surfaces of their contact
with the medium. Obviously, the growth of
the interphase surface affects the increase
in the rate of oxygen dissolution, but the
reduction of pressure according to Henry's
law limits the effectiveness of the process.
Returning to the condition (20) and
taking into account (21), we have:
.
у, m/s
τ, s
Figure 1. Diagram of changes in the
rising rate of gas bubbles in the
transient process
1...3 s τ
(f)
.
у
(f)
≈0,25...0,27 m/s
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