Biomedical Engineering Reference
In-Depth Information
water can be identified. Many surfactants added to water in a very small (ppm)
concentration, sufficiently render constant bubble terminal velocity at a given size.
A further increase in concentration produces no effect on the bubble terminal veloc-
ity. Grossly contaminated water represents the case of minimum constant terminal
velocity at a given size. The terminal velocity of small bubbles follows that of solid
spheres until the volume-equivalent Reynolds number exceeds a critical value. For
bubbles in pure and grossly contaminated water, the critical value is 25 and 60 re-
spectively. However, the bubble deformation is insignificant and the bubble shape
remains spherical up to a much higher Reynolds number which is about 200 [10].
For small bubbles (with
Re <
130) in contaminated water, the shape of the bubble
is spherical and the bubble rise velocity follows that of solid particles, i.e.,
U
Stokes
1
−
1
Ar/
96
U
=
+
,
(2)
0
.
079
Ar
0
.
749
)
0
.
755
(
1
+
8
R
b
δ
2
g/μ
2
is the Archimedes number for bubble rise.
The drag coefficient of bubble rise in contaminated water starts to deviate from
the standard curve of solid spheres at approximately
Re
where
Ar
=
130. At this regime, the
bubbles are non-spherical. The shape factor has to be considered in predicting the
drag coefficient of the bubble rise in contaminated water and the bubble terminal
rise velocity becomes a function of the Morton number,
Mo
=
gμ
4
/(δσ
3
)
where
σ
=
is the surface tension, as follows:
18
U
Stokes
4
a
2
Ar
2
b
−
1
Mo
0
.
46
b
2
.
85
1
2
−
2
b
U
=
,
(3)
where
a
and
b
are the model parameters [53]. The contaminated water is practi-
cally justified by the simple criterion of Clift
et al.
[10]: the velocity-volume curve
of bubbles should not pass through any maximum peak. This condition is usually
satisfied by the bubble rise in many surfactant solutions.
C. Deformation of Bubble Interacting with the Surface
When subjected to external forces, gas bubbles can be deformed and undergo dif-
ferent shapes. For freely rising bubbles before interacting with the solid surface,
the extensive experimental dada for the radius,
L
, of the bubble cross section area
projected perpendicularly to its path can be related to the bubble sphere-equivalent
radius,
R
b
, as follows [53]:
0
.
042
Ta
1
.
908
R
b
1
.
488
+
L
=
0
.
025
Ta
1
.
908
,
(4)
+
1
.
488
Re(Mo)
0
.
23
where
Ta
=
is the Tadaki number given as a function of the bubble
Reynolds number,
Re
2
R
b
Uδ/μ
, determined by the sphere-equivalent bubble ra-
dius,
R
b
, and the Morton number,
Mo
.
=
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