Materials Transactions, Vol. 45, No. 3 (2004) pp. 870 to 876
#2004 The Japan Institute of Metals
Filament and Droplets Formed Behind a Solid Sphere Rising Across
a Liquid-Liquid Interface
Momoko Abe1 and Manabu Iguchi2
1
2
Undergraduate student, Department of Materials Engineering, Faculty of Engineering, Hokkaido University, Sappporo 060-8628, Japan
Division of Materials Science and Engineering, Graduate School of Engineering, Hokkaido University, Sappporo 060-8628, Japan
An understanding of the dynamic behavior of non-metallic inclusions such as bubbles and alumina passing through an interface between
molten steel and slag is of essential importance for producing clean steel. Model experiments were carried out in this study using water and
silicone oil as the working fluids. The behavior of a solid sphere rising through an interface between stratified two liquid layers and the associated
deformation of the interface were observed with a high-speed video camera. A filament-like column of the lower liquid was formed behind the
sphere rising in the upper liquid layer. Many droplets were generated due to breakup of the column. Empirical equations were proposed for
parameters characterizing the shape and size of the column and droplets.
(Received October 24, 2003; Accepted January 13, 2004)
Keywords: steelmaking, inclusion, sphere, interfacial phenomena, filament, droplet
1.
Introduction
The dynamic behavior of non-metallic inclusions such as
bubbles and alumina in molten steel in the refining processes
is closely associated with the quality of the steel products.1,2)
Many investigations have been carried out to understand the
motions of non-metallic inclusions around an interface
between molten metal and slag. The previous investigations,
however, are mainly concerned with a liquid drop resting at a
liquid-liquid interface,3,4) a bubble rising through the interface5–9) and a solid sphere falling through the interface.10–13)
Information on a solid sphere rising through the interface is
very limited, although this is the case in the real refining
processes.
In this study attention was paid to a solid sphere rising
through an interface between silicone oil and water. Primary
concern was the effects of the kinematic viscosity of silicone
oil on the motion of the sphere and generation of water
droplets in the upper silicone oil layer. A high-speed video
camera was used for the observation of these phenomena.
Empirical equations were proposed for some parameters
characterizing the deformation of the interface and the water
droplets.
2.
Experimental Apparatus and Procedure
had an inner diameter, D, of 200 mm and a height, H, of
400 mm. Water was filled to a depth, HL , of 280 mm and
silicone oil was placed on the water layer. The thickness of
the silicone oil layer, Hs , was 110 mm. Three kinds of
silicone oil of different kinematic viscosities were used. The
density, s , and the kinematic viscosity, vs , were 0.936 g/cm3
and 10 mm2 /s, 0.965 g/cm3 and 100 mm2 /s, and 0.970 g/
cm3 and 1000 mm2 /s, respectively, as shown in Table 1. The
interfacial tension, ws , was approximately 53 mN/m for the
three cases. A sphere was made of polypropylene whose
density, p , was 0.849 g/cm3 . The sphere was wetted both by
water and silicone oil. The diameter of the sphere, dp , was
0.953, 1.27, 1.59, 1.91, 2.22, and 2.54 cm. The vessel was
enclosed with another vessel of a square cross-section, and
water was filled between the two vessels in order to reduce
the distortion of video images.
Each sphere was held with a cramp in a container settled
on the bottom wall of the vessel and then released. The sphere
rose along the guide pipe and issued into the bath. Time, t,
was measured from the moment at which the sphere left the
guide pipe, and the vertical distance, z, was measured from
the exit of the guide pipe. The behavior of a solid sphere and
deformation of the interface were observed with a high-speed
video camera at 500 frames per second and the images were
stored on a personal computer.
Figure 1 shows a schematic of the experimental apparatus.
The cylindrical test vessel made of transparent acrylic resin
Table 1
Sphere
Density (g/cm3 )
Diameter (cm)
Polypropylene
0.849
0.953, 1.27, 1.59
1.91, 2.22, 2.54
Liquid
Water
Fig. 1
Experimental apparatus.
Physical properties (at 298 K).
Density
(g/cm3 )
0.998
Kinematic
Interfacial
viscosity
tension
(cSt, mm2 /s)
(mN/m)
0.898
Silicone Oil 10
0.936
10
52.7
Silicone Oil 100
0.965
100
53.0
Silicone Oil 1000
0.970
1000
53.0
Filament and Droplets Formed Behind a Solid Sphere Rising Across a Liquid-Liquid Interface
Fig. 2 The axial position of a sphere with respect to time.
871
Fig. 4 Measured values of the drag coefficient in water against the
Reynolds number.
CD ¼ 4ðw p Þgdp =ð3w vp,tw 2 Þ
ð1Þ
where w and p are the densities of water and particle,
respectively, g is the acceleration due to gravity, dp is the
particle diameter, and vp,tw is the terminal velocity of particle
in the water layer.
The measured values of CD were compared with the
following empirical equation proposed by Rumph14) in Fig. 4.
CD ¼ 0:5 þ 24=Re
Re ¼ vp,tw dp =vw
ðRe < 105 Þ
ð2Þ
ð3Þ
where Re is the Reynolds number, vw is the kinematic
viscosity of water. An agreement between the measured and
calculated values is satisfactorily good.
3.1.3 Drag coefficient in silicone oil layer
Figure 5 shows the measured values of the drag coefficient
of a sphere rising in the upper silicone oil layer. The solid line
indicates the value calculated from
Fig. 3
3.
The axial velocity of a sphere with respect to time.
CD ¼ ð24=ReÞð1 þ 0:15Re0:687 Þ
Re ¼ vp,ts dp =vs
ðRe < 800Þ
ð4Þ
ð5Þ
Experimental Results and Discussion
3.1
Drag coefficient of a sphere rising in water and
silicone oil layers
3.1.1 Position of rising sphere with respect to time
Figure 2 shows the vertical position, z, of a sphere for three
kinds of silicone oils. The velocity of the sphere was
determined by graphically differentiating the vertical position of the sphere with respect to time, t. The velocity of the
sphere shown in Fig. 3 reached once a terminal velocity
before arriving at a liquid-liquid interface and then reached
another terminal velocity before arriving at the free surface of
the silicone oil layer.
3.1.2 Drag coefficient in the lower water layer
The sphere rose following a zigzag path in the water layer,
although the amplitude of the path was small. The drag
coefficient CD was calculated from the following equation
obtained by equating the buoyancy force acting on the sphere
to the sum of the gravitational force and the hydrodynamic
drag.
Fig. 5 Measured values of the drag coefficient in silicone oil against the
Reynolds number.
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M. Abe and M. Iguchi
This equation was proposed by Schiller and Nauman.14) A
good agreement can be seen between the measured values
and eq. (4), supporting that the accuracy of the present
velocity measurement method is satisfactory.
3.2
Behavior of water column pulled up into the upper
silicone oil layer
A solid sphere arrived at a water-silicone oil interface at a
terminal velocity and entered into the upper silicone oil layer
accompanying water behind it (see Fig. 6). In the figure Nd is
the number of droplets. A water column therefore was
formed behind the sphere, as can be seen in Figs. 7 and 8.
Fig. 6 Sketch for the explanation of droplets formation process and the
explanation of definitions of Hm and other quantities.
Fig. 7 Image of droplet formation (vs =10cSt).
Fig. 8
Image of droplet formation (dp ¼ 2:22 cm).
Filament and Droplets Formed Behind a Solid Sphere Rising Across a Liquid-Liquid Interface
Fig. 9
873
Pattern of droplet formation process.
Figure 9 shows the existence of three types of water
column.
(1) Case A: the shape of the water column was complex and
large water droplets were generated.
(2) Case B: a filament-like water column was formed and
many small water droplets were generated due to a
hydrodynamic instability of the column.
(3) Case C: a water column was pulled back into the water
layer but no water droplet was generated.
More experimental results are required for identifying the
region where the three types of water column appear.
Reiter and Schwerdtfeger5,6) observed the behavior of a
liquid column formed behind a bubble rising across a liquidliquid interface. Some of the columns are similar to those
observed in this study.
3.3 Shape and size of water filament and water droplets
In the following, empirical equations will be proposed for
some parameters characterizing the sizes of water column
and water droplets.
3.3.1 Maximum filament length, Hm
The maximum filament length, Hm , is shown against the
particle diameter, dp , in Fig. 10. It is evident that Hm
increases with dp . The effect of the kinematic viscosity of
silicone oil, vs , on Hm is not so significant, although vs was
changed over a wide range. As the density difference between
water and silicone oil, (w s ), decreases with an increase
in vs , the change in Hm is attributable to the density
difference. A modified Weber number therefore was introduced to correlate Hm . The measured values of Hm can be
approximated by the following empirical equation within a
scatter of 30%, as shown in Fig. 11.
Hm =dp ¼ 3:09ðWew,s Þ0:80
ð6Þ
2
Wew,s ¼ ½ðw s Þdp vp,ws =ws vp,ws ¼ vp,tw þ vp,ts
1=2
ð7Þ
ð8Þ
Fig. 10 Maximum filament length of lower phase pulled into upper phase.
where Wew,s is the modified Weber number, dp is the
diameter of the sphere, w is the density of water, s is the
density of silicone oil, and vp,ws is the mean value of the
terminal velocities in the water and silicone oil layers, vp,tw
and vp,ts . This Weber number was introduced because the
motion of a filament was considered to be governed both by
the terminal velocities of a solid sphere in the water and
silicone oil layers.
3.3.2 Time required for a filament to reach the maximum height, tm
Figure 12 shows that the time required for a filament to
reach the maximum height, tm , is a decreasing function of dp
and (w s ). The measured values of tm can be approximated by
vp,ws tm =Hm ¼ 1:95ðWew,s Þ2:20
ð9Þ
A scatter of data points was within 50%, as shown in
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M. Abe and M. Iguchi
Fig. 13 Relation between vp,ws tm =Hm and Wew,s .
Fig. 11 Relation between Hm =dp and Wew,s .
Fig. 12 Relation between tm and dp .
Fig. 14 Total volume of water droplets in silicone oil layer.
Fig. 13. Such a scatter is acceptable in this kind of
measurement.
3.3.3 Total volume of water droplets, Vt
Figure 14 shows that the total volume, Vt , is an increasing
function of the sphere volume, dp 3 =6, and density difference, (w s ). The following empirical equation could
approximate the measured values of the total volume of water
droplets within a scatter of 50% (see Fig. 15).
Vt =ðHm dp 2 Þ ¼ 0:0315ðWew Þ3:01
Wew ¼ ½ðw 2
s Þdp vp,tw
=ws 1=2
ð10Þ
ð11Þ
where Wew is another type of Weber number and vp,tw is the
terminal velocity in the water layer. This Weber number was
introduced because the amount of water carried into the
silicone oil layer by a sphere was considered to be governed
mainly by the terminal velocity of a solid sphere in the water
layer.
Reiter and Schwerdtfeger5,6) proposed an empirical equation for the total volume of water droplets formed behind a
bubble in addition to those for the average residence time of
Fig. 15 Relation between Vt =ðdp 2 Hm Þ and Wew .
Filament and Droplets Formed Behind a Solid Sphere Rising Across a Liquid-Liquid Interface
875
Fig. 16 Total surface area of droplets in silicomne oil layer.
Fig. 18 Sketch for explanation of filament diameter and droplet diameter.
Fig. 17 Relation between St =ðdp Hm Þ and Wew .
Fig. 19 Relation between da and a.
the droplet in the upper layer and the average droplet
diameter. The equation thus derived for Vt could not predict
the presently measured values, although the evidence was not
shown here.
3.3.4 Total surface area of water droplets, St
The total surface area of water droplets, St , is shown in
Fig. 16. The measured values can be predicted by
St =ðHm dp Þ ¼ 0:513ðWew Þ1:89
ð12Þ
within a scatter of 40%, as shown in Fig. 17.
3.3.5 Relationship between mean diameter of water
filament, a, and mean diameter of water droplets
The water filament just before breakup into droplets is
assumed to be cylindrically shaped, as shown in Fig. 18. Its
height and diameter are represented by Hm and a, respectively. As the volume of the cylinder is equal to the total
volume of the water droplets, Vt , we have
a ¼ ½4Vt =ðHm Þ1=2
ð13Þ
Figure 19 shows the relationship between the mean diameter
of water droplets, da , and the mean diameter of the water
filament, a. The measured values of da were approximated by
the solid line expressed by
da ¼ 0:982a þ 0:126
ðcmÞ
ð14Þ
When a is smaller than approximately 0.4 cm, the measured
values can also be predicted by the following equation
proposed by Rayleigh based on an instability theory.15)
da ¼ 1:89a
ð15Þ
In this study the measurements were carried out for a
sphere wetted both by water and silicone oil. The wettability
of the sphere would affect the deformation of the interface
and the resultant formation of a filament and droplets. Further
experimental study is required for a full understanding of the
wettability effect.
4.
Conclusions
An interface between stratified water and silicone oil
layers was deformed by a solid sphere rising across it. The
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M. Abe and M. Iguchi
deformation of the interface and the generation of water
droplets in the upper silicone oil layer were observed with a
high-speed video camera. The deformation patterns were
classified into three types with respect to the diameter of the
sphere and the density difference. Water droplets were
generated in the silicone oil layer due to breakup of a water
filament formed behind the sphere. Empirical equations were
proposed for some parameters characterizing the water
filament and water droplets. These parameters are the
maximum filament length, Hm , time required for the filament
to reach the maximum height, tm , total volume of water
droplets, Vt , total surface area of water droplets, St , and mean
diameter of water droplets.
Nomenclatures
a: mean diameter of filament
da : mean diameter of water droplets
dp : diameter of solid sphere
Hm : maximum filament height
Nd number of droplets
St : total surface area of water droplets
tm : time required for a filament to reach the maximum
height
Vt : total volume of water droplets
vp,ts , vp,tw : terminal velocities of particle in the silicone oil
and water, respectively.
Wew : modified Weber number based on the terminal
velocity in water
Wew,s : modified Weber number based on the mean value of
the terminal velocities in water and silicone oil
vs , vw : kinematic viscosities of silicone oil and water,
respectively.
p , s , w : densities of particle, silicone oil, and water,
respectively.
ws : interfacial tension
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