Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 4.2.3 - Terminal velocity of a single drop in a vertical pipe in clean and fully-contaminated systems
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00103
 Material Information
Title: 4.2.3 - Terminal velocity of a single drop in a vertical pipe in clean and fully-contaminated systems Particle Bubble and Drop Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Kurimoto, R.
Hayashi, A.
Tomiyama, A.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: drag force
Taylor bubble
terminal velocity
contaminated system
 Notes
Abstract: Terminal velocities and shapes of drops rising through vertical pipes in clean and fully-contaminated systems are measured by using a high-speed video camera and an image processing method. Silicon oils and glycerol water solutions are used for the dispersed and continuous phases, respectively, and Triton X-100 is used for surfactants. Clean and contaminated drops take either spherical, spheroidal or deformed spheroidal shapes when the diameter ratio λ is less than a certain critical value, λC, whereas they take bullet shapes for λ > λC (Taylor drops). The validity of available drag and Froude number correlations is examined through comparisons with the measured data. The effects of surfactants are also discussed. The conclusions obtained are as follows: (1) λC strongly depends on the Eötvös number Eo, whereas the effects of the Morton number, M, and the viscosity ratio, μ*, on λC are relatively small, (2) we can obtain good evaluations of the terminal velocities of clean drops at any λ by selecting appropriate drag and Froude number correlations, (3) the terminal velocities of contaminated drops are well evaluated by taking the limit μ* → ∞ in the drag correlation for clean drops in the viscous force dominant regime, (4) the reduction in surface tension due to the addition of surfactants is the main cause of the increase in the terminal velocity and the decrease in the radius of a Taylor drop, and (5) the terminal velocity of a Taylor drop at high EoD and low μ* is not affected by surfactants, and therefore, the Froude number correlation for clean Taylor drops is also applicable to contaminated drops.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00103
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 423-Kurimoto-ICMF2010.pdf

Full Text

Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Terminal velocity of a single drop in a vertical pipe
in clean and fully-contaminated systems


Ryo Kurimoto, Kosuke Hayashi and Akio Tomiyama

Graduate School of Engineering, Kobe University
1-1, Rokkodai, Nada, Kobe, Japan
kurimoto@cfrg.scitec.kobe-u.ac.jp, hayashi@mech.kobe-u.ac.jp, tomiyama@mech.kobe-u.ac.jp


Keywords: Drag force, Taylor bubble, Terminal velocity, Contaminated system




Abstract

Terminal velocities and shapes of drops rising through vertical pipes in clean and fully-contaminated systems are measured
by using a high-speed video camera and an image processing method. Silicon oils and glycerol water solutions are used for the
dispersed and continuous phases, respectively, and Triton X-100 is used for surfactants. Clean and contaminated drops take
either spherical, spheroidal or deformed spheroidal shapes when the diameter ratio k is less than a certain critical value, Xc,
whereas they take bullet shapes for X > kc (Taylor drops). The validity of available drag and Froude number correlations is
examined through comparisons with the measured data. The effects of surfactants are also discussed. The conclusions obtained
are as follows: (1) kc strongly depends on the Edtvos number Eo, whereas the effects of the Morton number, M, and the
viscosity ratio, L*, on Xc are relatively small, (2) we can obtain good evaluations of the terminal velocities of clean drops at
any X by selecting appropriate drag and Froude number correlations, (3) the terminal velocities of contaminated drops are well
evaluated by taking the limit g* co in the drag correlation for clean drops in the viscous force dominant regime, (4) the
reduction in surface tension due to the addition of surfactants is the main cause of the increase in the terminal velocity and the
decrease in the radius of a Taylor drop, and (5) the terminal velocity of a Taylor drop at high EoD and low g* is not affected by
surfactants, and therefore, the Froude number correlation for clean Taylor drops is also applicable to contaminated drops.


Introduction

The CMFD (Computational Multiphase Flow Dynamics)
now plays an important role in designing various industrial
devices. Among various CMFD methods, multi-fluid models
and Eulerian-Lagrangian methods are appropriate for dealing
with a huge number of bubbles and drops in practical devices.
These methods, however, require reliable closure relations
for forces acting on fluid particles. Since the motion of a
fluid particle is affected by the presence of walls, the effects
of walls must be taken into account in the closure relations.
However, our knowledge on the wall effect is still
insufficient even for fluid particles rising through stagnant
liquids in vertical pipes.
The dynamics of a fluid particle rising through an infinite
stagnant liquid is governed by the five forces, i.e., the
viscous forces, Fc and FD, of the continuous and dispersed
phases, the surface tension force F,, the inertial force F,, and
the buoyancy force F,. Relevant dimensionless groups can be
represented by using these forces, e.g., the Reynolds number,
Reo, of a free-rising fluid particle, the E6tvds number, Eo,
and the viscosity ratio, t*:

17 pcT70d
Reo F c (1)
F c F c


Eo Fb Apgd2
Eo=
F, a


FD _LD
FMc -c


where p is the density, VTo the terminal velocity in an infinite
stagnant liquid, d the sphere-volume equivalent diameter of a
fluid particle, g the viscosity, Ap the density difference
between the two phases, g the magnitude of the acceleration
of gravity, and c the surface tension. The subscripts C and D
denote the continuous and dispersed phases, respectively.
The drag coefficient, CDO, of a fluid particle in the infinite
stagnant liquid is given as a function of these dimensionless
groups. Win Myint (Win Myint et al., 2006; Win Myint,
2008) proposed the following drag correlation for drops in
clean systems:


CDO = max[CDO,, CDOo,


where CDo0 and CDOo are the drag coefficients in the viscous
force dominant (g) and surface tension force dominant (a)
regimes, respectively. The CDO, is a combination of the
Hadamard-Rybczynski solution for fluid particles in Stokes
flows (Hadamard, 1911; Rybczynski, 1911) and the inertial





Paper No


effect multiplier K, (Schiller & Naumann, 1933):

_CDO 8 (2+3Lt* KJ
Reo 1+g* )


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

correlation using experimental data in literature (Salami et al.,
1965; Strom & Kintner, 1958; Uno & Kintner, 1956):


CDo = CDOoKW


where Kw is the wall effect multiplier given by


where Ki, is given by


KI, = 1+0.15ReO 687 (6)

The drag coefficient, CD0o, is given by

8 Eo
CDO0 (7)
3 Eo+4

This model is based on a wave analogy (Mendelsen, 1967;
Tomiyama et al., 1998). Taking the limit Eo o in Eq. (7)
gives the following drag correlation in the inertial force
dominant (i) regime:

8
CDO, (8)
3

Equation (4) indicates that ut* affects VT0 in the ug regime,
whereas VT0 is independent of ug* in the C and i regimes.
Shapes of fluid particles are spherical, deformed spheroidal
and spherical-cap in the ug, C and i regimes, respectively.
Since the shape and terminal velocity, Vr, of a single fluid
particle rising through a stagnant liquid in a vertical pipe
depend on the ratio, L (= d / D), of the sphere-volume
equivalent diameter d to the pipe diameter D, the effects of L
must be implemented into drag correlations. When L is less
than a certain critical value Lc, Vr varies with L. Fluid
particles in a pipe for 0 < L < kc take either spherical,
spheroidal or deformed spheroidal shapes due to the wall
effect, even if the viscous force is dominant. In this case, the
following drag correlation (Hayashi & Tomiyama, 2009) can
be used for fluid particles in clean systems:


_ 8 ( 2+31*) 1.
Re 1+u.* )


where CD, is the drag coefficient, Re the Reynolds number of
a fluid particle in a pipe, and Kwh the wall effect multiplier
(Haberman & Sayre, 1958), which are defined by


Re pcVd
K tc


( CL2+3*)


Kw = (1-12)-3


Equation (12) is valid for 0 < L < 0.6. Since the database they
used includes only bubbles and drops in the C regime, their
correlation might be applicable only to this regime.
The wall effect on the terminal velocity of a spherical-cap
bubble in the i regime was investigated by Collins (1967),
and Wallis (1969) proposed a correlation based on Collins'
data for 0.125 < < 0.6:


CD, = [0.78exp(2)l]CDo,


Bubbles take bullet shapes for L > kc. They are called
Taylor bubbles. The terminal velocity of the Taylor bubble is
independent of 1, which means that the characteristic length
governing Vr is not d but D. Many studies have been carried
out on Taylor bubbles. Correlations proposed by Viana et al.
(2003) and Hayashi et al. (2010a) give good evaluations of
VT of Taylor bubbles for a wide range of fluid properties. The
latter correlation is given by

S-4463
0.0089 41
FrD0.0089 -1+ 4 9 (15)
F 0.0725+- (-0.11Re 33) oD 9
ReD D

where FrD is the Froude number, ReD the Reynolds number,
and EoD the E6tv6s number defined by


FrD VT
ApgD / pc


SApgD2
EOD -


pcVrD
ReD


(10) Drops also take bullet shapes for 2 > kc. Since their shapes
are similar to Taylor bubbles, they can be also referred to as
Taylor drops. Hayashi et al. (2010b) have recently proposed
a FrD correlation for Taylor drops by implementing the
effects of the viscosity ratio into Eq. (15):


1+c 2+3c k+c12 +* 3L +C3 2-53L*) +C4 _i+,61
1+u.* 1+u.* 1+u* 1+* )
(11)

where co = 2.2757, cl = -0.7017, c2 = 0.20865, c3 = 0.5689
and c4 = -0.72603. Equation (9) is applicable for 0 < L < 0.6.
Clift et al. (1978) proposed the following simple CD


1 1+1.75 11Re 33
0.0725D+ 1(. 0. lRe33
ReD 1+0.27ut*J


This equation also gives good evaluations of FrD for a wide






Paper No


range of fluid properties.
Taylor bubbles and Taylor drops can be also classified into
several regimes from the point of view of the dominant
forces (White & Beardmore, 1962), e.g., the viscous force
dominant (low ReD, high EoD), the inertial force dominant
(high ReD, high EoD), and the surface tension force dominant
(low EoD) regimes. Equations (15) and (19) are applicable to
all the regimes (Hayashi et al., 2010a, 2010b). Since this
classification is not essential in this study, we do not make
further discussion on this point.
It is possible to evaluate Vr for 0 < X < 0.6 and X > Xc by
using the above correlations. The critical diameter ratio,
however, depends on fluid properties and the pipe diameter,
e.g., kc is about 0.6 for an air bubble rising through water in
a pipe of D = 24.8 mm (Nakahara & Tomiyama, 2003),
whereas Xc 1 for bubbles and drops at low Reynolds
numbers (Coutanceau & Texier, 1986, D = 35 mm;
Almatroushi & Borhan, 2004, D = 8 mm). Hence the effects
of the relevant dimensionless groups on Xc and the
applicability of the above correlations to the range of 0.6 < X
< Xc should be examined.
In this study, terminal velocities of drops in vertical pipes
in clean systems are, therefore, experimentally investigated
to examine the applicability of the available correlations to
drops in pipes for 0.6 < X < c. The Vr correlations described
above are applicable only to clean fluid particles in pipes.
However multiphase systems in industrial devices may be
contaminated in some cases. In spite of its great importance,
there are few studies on the terminal velocity of a
contaminated fluid particle in a vertical pipe. Almatroushi &
Borhan (2' '14) investigated the effects of surfactants on Vr of
bubbles and drops in a vertical pipe. They confirmed that the
addition of surfactants decreases Vr for k < Xc, whereas VT
increases with the surfactant concentration for X > Xc. They
concluded that the Marangoni effect and the reduction in
surface tension are the main causes of the change in VT for X
< Xc and for X > Xc, respectively. However they dealt with
only fluid particles in Stokes flows, and therefore, little has
been known on the effects of the surfactants on fluid particles
at higher Reynolds numbers.
Experiments of drops in fully-contaminated systems in
pipes are, therefore, also carried out to investigate the effects
of surfactants on VT.

Experiments

The experimental apparatus is shown in Fig. 1. It consists
of the vertical pipe and the square tanks. Two pipes of D = 11
and 21 mm were used. The pipe length was 1,530 mm. The
pipes and the tanks were made of transparent acrylic resin.
The experiments were carried out at atmospheric pressure
and room temperature (the temperature of the liquid and
drops were kept at 298 + 0.5 K).
Silicon oils of various viscosities (Sin-etsu Silicon,
KF96-10, 30, 100, 300, and 500) and glycerol-water
solutions were used for the dispersed and continuous phases,
respectively. Triton X-100 (CH17C6H4(OCH2CH 2)1oOH) was
used for surfactants. The concentration of Triton X-100 was
1.0 mol/m3 at which the surface tension reached the value at
the critical micelle concentration as shown in Fig. 2. The
density, viscosity, surface tension and temperature were
measured using a densimeter (Ando Keiki Co., Ltd., JIS


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

B7525), a viscometer (Rion Co., Ltd., Viscotester VT-03E),
capillary tubes (glass tube, 1.02 mm i.d.) and a digital
thermometer (Sato Keiryoki MFG Co., Ltd., SK-1250MC),
respectively. Fluid properties were measured at least five
times. Uncertainties estimated at 95 % confidence in
measured p, and a were 0.021, 0.53 and 4.0 %,
respectively. The measured values of these fluid properties
agreed well with the data in literature (Ishikawa, 1968). The
values given in literature were, therefore, used in calculations
of the dimensionless numbers. Fluid systems examined are
summarized in Table 1, in which M is the Morton number
defined by


4
l g=cAp
M= 2 3
23
Pc0


A small amount of silicon oil was stored in the
hemispherical glass cup. A single drop was released by
rotating the cup. The visualization of the drop motion and
shape using a high-speed video camera (Integrated Design
Tools Inc., Motionscope M3, frame rate = 20 400 frame/s,
spatial resolution z 0.095 mm/pixel) confirmed that most of
drops were axisymmetric and rose rectilinearly along the
pipe axis. Several small drops, however, did not move along
the pipe axis. In addition, several Taylor drops did not take
stable bullet shapes even at large X due to the
Rayleigh-Taylor instability as shown in Fig. 3. This
instability occurs at high EoD and low M in both clean and
contaminated systems. Only the axisymmetric drops rising
rectilinearly along the pipe axis are discussed in this study.
Successive images of drops in the pipe were taken by
using the high-speed video camera at 1,300 mm above the
bottom of the pipe to obtain experimental data only at
terminal conditions. The pipe was enclosed with the acrylic
duct. The gap between the pipe and the duct was filled with
the same glycerol-water solution as that in the pipe to reduce
optical distortion of drop images. A LED light source
(Hayashi Watch-Works, LP-2820) was used for backward
illumination.


High-speed
video camera

Vertical pipe

Duct -








Lower tank


Side view


LED
light
source





1300mm


High-speed
video camera


Cross-sectional view
A-A'
Hemispherical
cup
Nozzle
Syringe


Figure 1: Experimental apparatus





7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Table 1: Experimental conditions
Case A B C D E F G H I
Pc [kg/m3] 1240 1240 1240 1219 1219 1219 1189 1189 1189
pD [kg/m3] 955 970 970 955 965 970 935 955 965
we [mPa-s] 262 262 262 85 85 85 25 25 25
D [mPa-s] 29 291 485 29 97 485 9.4 29 97
c [mN/m] 30 29 29 32 31 30 33 33 31
D [mm] 11, 21 11,21 11,21 11,21 11,21 11,21 21 21 21
Grycerol 93 93 93 85 85 85 74 74 74
Concentration [%]
Silicon oil, KF96- 30 300 500 30 100 500 10 30 100
logM -0.50 -0.48 -0.48 -2.6 -2.5 -2.5 -4.7 -4.8 -4.7
1t* 0.11 1.1 1.9 0.34 1.1 5.7 0.37 1.1 3.8
EoD 11, 41 11,40 11,40 9.8, 36 9.7, 35 9.8, 36 33 31 31


I I I I I I I I I I I I I I I
10 41 0 -2 0-0 1100 102 1010- 10 110010110210 0-3 0-2 100101102
Molar concentration [mol/m3]
Figure 2: Dependence of surface tension on surfactant
concentration















(a) Clean (b) Contaminated
EoD = 77 EoD = 36
Figure 3: Clean and contaminated Taylor drops under
unstable conditions (logM= -2.5 and g* = 5.7)

N ----- ,,,,,,,,,,,,,,,,,

( O-


(a) Original image %


(b) Binary image (c) Reconstructed
drop shape
Figure 4: Reconstruction of drop image


(a) d = 4.8 mm, = 0.44
almost spherical


(b)d= 9.2mm, = 0.83 (c) d= 15.6 mm, = 1.42
deformed spheroidal bullet shape (Taylor drop)
Figure 5: Drops in case A (D = 11 mm)

The velocity, Vr, the diameter, d, and shape of a drop were
calculated from the drop images using an image processing
method (Tomiyama et al., 2002). An example of original
drop images is shown in Fig. 4 (a). The drop image was
transformed into binary images using an appropriate
threshold level (Hosokawa & Tomiyama, 2003) as shown in
Fig. 4 (b). Since drops were axisymmetric, all the horizontal
cross-sections of a drop were circle of the radius R,, where
the index, i, denotes the pixel number in the vertical direction.
The height of a circular disk in the image was one pixel and
its physical length was Az. The resultant circular disks were
piled up in the vertical direction to reconstruct a
three-dimensional drop shape as shown in Fig. 4 (c). Hence
the sphere-volume equivalent diameter is given by
d = Y' (6RAz)l/3 where N is the total number of pixels
in the vertical direction. The maximum errors in measured
diameter were less than 1 %. The velocity of a drop was
computed from two binary images by measuring the
difference in elevation of the noses of the two drop images.
The measured VT showed that the drops reached their
terminal conditions at the measuring section.

Results and Discussion

Clean drops. Figure 5 shows examples of drop shapes in
the pipe of D = 11 mm in case A (logM = -0.50, EoD =11
and g* = 0.11). Drops for X < 0.5 are almost spherical (Fig. 5


Paper No


logM = -0.5'0
-*=0.11
-o 00


logM = -2.6
W*=0.34
0 0
0


logM = -4.7
W(50.37


O


00


00


n O






Paper No


(a)). Since the wall effect becomes significant with
increasing L, drops do not have the fore-aft symmetry for L >
0.5 as shown in Fig. 5 (b). Figure 5 (c) shows an example of
Taylor drops. As pointed out by Goldsmith & Mason (1962),
the nose and rear of the Taylor drop in the high viscosity
system are prolate spheroidal and oblate spheroidal,
respectively.
Figure 6 shows terminal velocities of drops at logM -
-0.49 (cases A, B and C). The curves in the figure are the
terminal velocities, VTO, of drops in the infinite stagnant
liquids evaluated by using Eq. (5) (Win Myint et al., 2006;
Win Myint, 2008). The VTr monotonously increases with d
and decreases with increasing pL*. In these cases, all the
drops in the infinite liquids are in the pg regime. The presence
of pipe wall decreases the terminal velocity, i.e., VT < Vro.
The terminal velocity, VT, varies with L for L < 1, whereas it
is independent of L for L > 1. Hence the drops for L > 1 are
Taylor drops. All the drops in the pipe depend on pg*, and
therefore, the viscous force is dominant not only for k < 1
but also for L > 1. There is a transition region from deformed
spheroidal drops to Taylor drops. As shown in Fig. 7, VT in
the transition region smoothly connects with those in the two
shape regimes.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0.015
VTO for
free drops t* Data Eq.(5)
r o 0 .1 1 o
1.1 A -.
1.9 ----
0.01 -
,0 Deformed
i spheroidal Taylor drop
I /e% Transition

0.005 0

(00 00 000


0 5 10 15 20
d [mm]
Figure 6: Terminal velocities of clean drops in a high
viscosity system (cases A, B and C: logMA -0.49,
EoD 10, D = 11 mm)
004 .


0.03



0.02



0.01


0 5 10 15 20 25
d [mm]
Figure 7: Transition from deformed spheroidal drops to
Taylor drops (case A, logM= -0.50, L* = 0.11)


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Examples of drops in the pipe of D = 21 mm in case H
(logM = -4.8, EoD = 31 and ug* = 1.1) are shown in Fig. 8.
The drops for k < 1c are more deformed than those in Figs. 5
(a) and (b). The nose of the Taylor drop shown in Fig. 8 (c) is
almost spherical and the rear is flatter than that of Fig. 5 (c).
At logM -4.7 (cases G, H and I), drops in infinite
stagnant liquids suddenly transit from the ug regime to c and i
regimes as shown in Fig. 9. The tendency of VT at logM -
-4.7 is similar to that at logM_ -0.49. However the effect of
ug* on VT is very small for 0.5 < k < 0.6. The surface tension
force is, therefore, dominant in this range. The Taylor drops,
however, depend on ug*, and therefore, the effects of ug* on
VT reappear in the transition region.
Measured terminal velocities in several cases are
compared in Fig. 10. Experimental data of air bubbles in
water obtained by Nakahara & Tomiyama (2003) are also
plotted in the figure. The comparison indicates that the
critical diameter ratio, at which VT is equal to the VT of
Taylor drops, depends on the dimensionless groups, EoD, M
and g*. The effect of EoD on cL is large but those of M and
ug* are small. The critical diameter ratio can therefore be well
correlated in terms of EoD as shown in Fig. 11.


(a) d = 5.3 mm, h = 0.25
deformed spheroidal


(b)d= 11.3 mm, = 0.54 (c) d= 25.8 mm, = 1.23
deformed spheroidal bullet shape (Taylor drop)
Figure 8: Drops in case H (D = 21 mm)


0 0.2 0.4 0.6 0.8 1 1.2 1.4
0.2
VTO for free rising drops

0.15- a & i-regimes
E.7) Data Eq.(4)
7 -regime 1.17
0.1- 3.8 D ---



0.05 n
.Transition
Deformed spheroidal Taylor drop
0 10 20 30
d [mm]
Figure 9: Terminal velocities of clean drops in a low
viscosity system (cases G, H and I: logMA -4.7,
EoD, 32, D= 21 mm)


0 EOD = 41 (D=21 mm)
SEOD = 11 (D=11 mm)

o 0b ooo C O CDO Q
cP
(9 Deformed "-_ -0
O spheroidal Tranlslon Taylor dr(
0
O

reformed
~e .r..-.r:,. l31Trans ol" Taylor drop


L^ tNAA






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


10-1 100
Eo


101 102


Figure 10: Comparison of critical diameter ratio, Xc

14.


0.8


0n


Ai


S'0
0


20 40 60 80 100
EoD
Figure 11: Xc vs. EoD


Figure 12 shows measured terminal velocities in case B
(logM= -0.47, i* = 1.1, EoD = 12, and Xc 1) on the Re-Eo
plane. The Re is less than unity, and therefore, Eq. (9) can be
used at least for X < 0.6. The terminal velocities of Taylor
drops can be evaluated using Eq. (19). The experimental data
and Eqs. (9) and (19) are compared in Fig. 12. Equation (9)
gives reasonable evaluations not only for X < 0.6 but also for
0.6 < 2 < kc. Equation (19) also agrees well with the
measured data.
Measured Reynolds numbers in case D (logM= -2.6, EoD
= 36, it* = 0.34, and 2c 0.8) are shown in Fig. 13. For
drops in the infinite liquid, the surface tension force is
dominant for Eo > 10 and the drag coefficient, CDo, is given
as a function of Eo. On the other hand, drops in the pipe are
still in the ug regime even for Eo > 10 because of the velocity
reduction by the wall effect. Equation (9) and the data are in
good agreement for k < kc. Hence Eq. (9) is applicable to
drops in the ua regime even for 0.6 < X < kc.
As mentioned above, drops in case H (logM= -4.7, EoD
31, u.* = 1.1, and 2c 0.8) are either in the ug regime or Ca
regime for X < 2c. The sudden transition from the former to


Figure 12: Comparison between calculated and measured Re
in case B (logM= -0.48, EoD = 11 and u* = 1.1)


r 100
<" in


10-1 100 101 102
Eo
Figure 13: Comparison between calculated and measured Re
in case D (logA= -2.6, EoD = 36 and u* = 0.34)

the latter occurs at X 0.3 as shown in Fig. 14. Equation (9)
is no longer applicable to the drops in the surface tension
force dominant regime, but Eqs. (7) and (13) might be
applicable. The drag correlation of drops in these regimes
can, therefore, be given by


CD = max --2+3t* [Kwh + K ,
LRe W 1+t* )


8 Eo K] (21)
], Eo+4
3 Eo + 4


The second term in the bracket, CDo,(Eo)KcW(t), which is for
drops in the C regime, does not include ug*. This is consistent
with the experimental result shown in Fig. 9. Equation (21) is


Paper No


100




10-1




S10-2



10"3


4 I -4


/ Eq. (9)
-- (Hayashi & Tomiyama, 2009)-
----- Eq. (4)
(Win Myint et al., 2006)
S -----Eq. (19)
/ (Hayashi et al., 2010b)
I....I...I


-o

06

oO


O Present
A Nakahara & Tomiyama (2003) (Bubble)
I I ,


I1


10-2






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


) 100


FCD(Eo,x)
Deformed spheroidal
CD(Re,W*,)


-Eq. (21) Present
-Eq. (9)
(Hayashi & Tomiyama, 2009)
---- Eq. (4)
(Win Myint, 2008)
----Eq. (19)
(Hayashi et al., 2010b)


I . .1


I . .1


Table 2: Velocity ratio, Rv
at lo M= -2.5


101
Co
(0
100

01


+


]10-3
10-2


Eo
Figure 14: Comparison between calculated and measured Re
in case H (log = -4.7, EoD = 31 and g* = 1.1)


V(contaminated) / V(clean),


EoD Case D Case E Case F
(*=0.34) (*=l.l) (g*=5.7)
9.8 1.24 1.59 2.59
36 1.01 1.08 1.18


EOD
Figure 16: Function G (=[l+41/Eo1 96]4 96)


0 0.i
0.08 I
Deformed
spheroidal


0.06



0.04



0 02


0 5 10 15
d [mm]


20 25 30


Figure 15: Terminal velocities of clean and
fully-contaminated drops in cases D and F (logA
-2.5 and EoD = 36)

compared with the data in Fig. 14. This correlation well
reproduces the measured Re for X < Xc.
Thus we could confirm that combining the available CD
correlations for clean drops allows us to reasonably evaluate
VT at any L.

Contaminated drops. Terminal velocities of clean and
fully-contaminated drops at log = -2.5 (cases D and F) are
compared in Fig. 15. All the clean drops for L < Xc depend
on g*, and therefore, they are in the ug regime. The
surfactants reduce VT of these drops. This is due to the
well-known Marangoni effect. The comparison of VT
between the contaminated drops in cases D and F indicates
that the viscosity ratio does not have much influence on VT
for L < 0.5 even though the drops are in the ug regime.


(a) clean (D) contamnnatea


Figure 17: Comparison between clean
drop shapes in a pipe of D
(logA = -2.6, L* = 0.34, EoD


and contaminated
21 mm in case D
=36, and L = 1.0)


The terminal velocities of the fully-contaminated drops are
higher than those of the clean drops for L > 1c. There is only
a slight change in Vr in case D, whereas the contaminated
Taylor drops are 20 % faster than the clean ones in case F. In
contrast to the contaminated drops for X < 0.5, the terminal
velocities of the contaminated Taylor drops depend on uP*.
The ratio, R7, of the terminal velocity of a
fully-contaminated Taylor drop to that of a clean drop at the
same values of the dimensionless numbers is summarized in
Table 2, which shows that EoD and ug* affect R7 at a constant
M. The influence of the surfactant is significant at low EoD
and high ug*. As pointed out by Almatroushi & Borhan
(2-'1'4), the EOtvos number in a contaminated system is
higher than that in the clean system because of the reduction
in surface tension (see Fig. 2). The factor, G =
[1+41/Eo' 96]-496, in Eq. (19) is drawn in Fig. 16. As can be
understood from Eq. (19), VT increases with G(EoD). As
shown in Fig. 16, the gradient, dG/dEoD, decreases with
increasing EoD, i.e., dG/dEoD is large at EoD = 9.8, whereas
small at EoD = 36. Hence the terminal velocities of Taylor
drops at small EoD are strongly affected by the surfactant.


Paper No


1 0 0 1
10-1
10-1


-- Taylor drop
Translation


ni^a-cr 'a HI +
,'. =0 34 (case D)
+ A ,lean
S+ ,-ontaminated

x xx Xx X
,o C9 o o 0 (
L'=' (case F)
0 Clean
X Contaminated
I I i I ,





7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Shapes of a clean and a contaminated Taylor drop in case
D (g* = 0.34 and Rv = 1.01) are compared in Fig. 17. The
rear shapes are different, whereas the nose shapes are the
same. This indicates that the surfactant concentration in the
nose region is low, while it is high in the rear part. The
reduction of surface tension is, therefore, negligibly small in
the nose region. This surfactant distribution at the interface
can be understood from the stagnant-cap model for spherical
bubbles (Griffith, 1960; 1962). If the interfacial convection is
much faster than the surfactant diffusion and the desorption,
adsorbed surfactants near the nose of a Taylor drop are
immediately transported toward the rear and they are
collected in the rear region. Since the viscosity ratio in case
D is low (ug* = 0.34), the interface mobility is high and the
surfactants are immediately transported toward the rear. Thus
the nose shape is independent of the presence of surfactants.
Let us discuss the main cause of the change in drop radius
in the axial direction. Figure 18 shows a schematic of the
contaminated drop. If the viscosity ratio is not so high, the
interfacial shear stress can be neglected. At the top and
bottom of the cylindrical section of the drop, the dynamic
equilibrium conditions are, therefore, given by the Laplace
formula:


Liqula Pipe



r PDPT=PT+T/RT
PD P

I

RB E P PB=P B+oB/RB

I P B/ 3 *^ --= -- ------
B_ D =PC a




Pipe axis
Figure 18: Schematic of contaminated Taylor drop


P T _pT
T
Sc

B
B = pB 0
PD-C + RB


where P is the pressure, R the radius of the cylindrical
section of the drop, and the superscripts T and B denote the
top and bottom regions of the cylindrical section,
respectively. Since we can postulate that PDB PDT P -
Pcr, we obtain


T RB
RT RB


Since the surfactant concentration at the interface is higher in
the bottom region, the surface tension CB is smaller than C,
i.e. c < CT. Hence


RB < RT


Figure 19 shows a clean and a contaminated drop in a high
Rv case (case F: gt* = 5.7 and Rv = 2.59). Their shapes are
different not only in the rear region but also in the nose
region. Figure 20 shows schematics of velocity profiles in the
radial direction in liquid films. The boundary condition for
the velocity at the interface is slip for clean Taylor drops of
ug* = 0 (Fig. 20 (a)), whereas the interfacial velocity
decreases with increasing PL* (Fig. 20 (b)). The low
interfacial velocity in the latter case allows surfactants to stay
long in the nose region. In addition, not only the increase in
viscosity ratio but also the surfactants decrease the interfacial
velocity. Consequently the surfactant may cover not only the
rear region but also the nose region, and therefore, the
reduction of the surface tension would occur in much wider
surface area including the upper part of the drop. Thus the
contaminated drop could be more slender than the clean one.


(a) clean (b) contaminated


Figure 19: Comparison between clean
drop shapes in a pipe of D
(logM= -2.5, ** = 5.7, EoD =


Liquid
film
Drop

-. T VTI
T 1

Pip g 4
wa ) *=
(a) g*=0


and contaminated
S11 mm in case F
9.8, and = 1.35)


Liquid
film
Drop
S VT




(b) u.* > 0


Figure 20: Velocity profiles in liquid films

We have also carried out interface tracking simulations of
clean and contaminated Taylor drops using our original
interface tracking method (Hayashi et al., 2006). The
computational grids and the boundary conditions were the
same as those used in Hayashi et al. (2010b). Figure 21 (a)
shows a comparison between measured and predicted drop
shapes corresponding to the one shown in Fig. 17 (a). The
shapes are in good agreement. In a simulation of the
contaminated Taylor drop shown in Fig. 17 (b), the
distribution of surface tension was varied as shown in Fig. 21
(b). This distribution is based on the above discussion, i.e.,
the concentration of surfactant is low in the nose region,


Paper No





Paper No


while it is high in the rear region. The tangential component
of surface tension force due to the Marangoni effect was also
accounted for. The other fluid properties were set at the
measured ones. The predicted drop shape agrees well with
the measured one. Figure 22 shows predicted drop shapes for
the drops in Figs. 19 (a) and (b). All the fluid properties were
set at the measured ones in Fig. 22 (a), whereas only the
surface tension was uniformly reduced all over the drop
interface in Figs. 22 (b) and (c). The drop length increases
with decreasing C. This is consistent with the above
discussion. The elongation is, however, too large when the
value of C at the critical miscelle concentration, 0.0023 N/m,
is used. The intermediate value of a, 0.017 N/m, gives a
more reasonable prediction. This indicates that the interface
is widely covered by the surfactant but its concentration at
the interface is lower than the critical miscelle concentration.
The rear shape in Figs. 22 (b) is, however, different from the
measured one. This discrepancy implies that the surfactant
concentration in the rear region is non-uniform in reality.
These simulations support the validity of the above-described
discussion on the effects of surfactants on shapes of Taylor
drops.
Let us briefly review available drag correlations for
contaminated drops in infinite stagnant liquids. Levich
(1962) implemented the effects of surfactants into the
Hadamard-Rybczynski solution for spherical drops:


CDO 8 (2+31t*+3C/c (26)
Reo 1+t*+C/tc (26)


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

where C is a coefficient expressing the retardation of
interface motion by surfactants. This equation was extended
by Win Myint et al. (2006) so as to be applicable to
non-spherical drops in the iL regime:


C 8 (2+3t*+3C/ tc
CDo Reo 1+t*+C/c )"


They measured Vo of fully-contaminated drops and
confirmed that Eq. (27) with the limit C co


CDO = 24 K
D Re Is,
Re0


agrees well with the measured data. Equation (28) is nothing
but a correlation for solid spheres (Schiller & Naumann,
1933), and therefore, the viscosity ratio does not affect VT of
fully-contaminated drops even in the it regime. Since we can
take the limit gt* co in Eq. (27) instead of C co to obtain
Eq. (28), drops of gt* co can be regarded as solid particles
from the point of view of the drag coefficient.
As shown in Fig. 15, the terminal velocities of the
fully-contaminated drops for X < kc do not depend on Pt*.
Hence it is expected that Eq. (9) is also applicable to the
fully-contaminated drops by taking the limit gt* co, i.e.


24 ,
CD = -[K2 (t* --, kX) + Kj 1]
Re


Measured Predicted











Case 1
clean


Case 2
contaminated


where


1-(co/3)15
K (* -> ,) )= I- (co /3) (30)
1+3C1 + c23 -3c3X5 C4X6

Equation (29) is compared with the data in case D in Fig. 23.
They are in good agreement up to X 0.5.


concentration


(a) (b)
Figure 21 Predicted drop shapes; left: clean (uniform a),
right: contaminated (non-uniform T)


A


(a) a = 0.030 (clean) (b) a = 0.017 (c) a = 0.0023 N/m
Figure 22 Predicted drop shapes for the drops in Fig. 19


I U '1
10-1
lu10-1


100 101 102


Eo
Figure 23: Reynolds numbers of drops in fully-contaminated
system compared with those in clean system in
case D (logM= -2.6, EoD = 36 and gL* = 0.34)


- I I I ' '
: logM=-2.6, g*=0.34, EOD=36
O Clean
A Fully-contamintated FrD(ReD,EoD,i*)
Eq.(29): Present Taylor drop
----- Eq.(19)
(Hayashi et al., 2010b)






0 CD(ReDe,)
Deformed spheroidal




/ 0 CD(RD X


. -I





Paper No


10-'
101


100 101


Eo
Figure 24: Reynolds numbers of drops in fully-contaminated
system compared with those in clean system in
case H (logM= -4.7, EoD = 31 and ug* = 1.1)

As shown in Table 2, the surfactant does not have much
influence on VT of Taylor drops at high EoD and low ug*. In
that case, Eq. (19) for Taylor drops in the clean system can
be also used for the drops in the fully-contaminated system.
Good agreements between Eq. (19) and the data of
contaminated Taylor drops were obtained as shown in Fig.
23.
Fully-contaminated drops in case H (logA = -4.7) are in
either the ug or c regime for L < kc as shown in Fig. 24. The
terminal velocities of the drops in the ug regime are well
evaluated by using Eq. (29). The drag coefficient, CDO, for
the contaminated drop in the c dominant regime, however,
needs to be studied in the future.

Conclusions

Terminal velocities, VT, and shapes of clean and
fully-contaminated drops in vertical pipes were
experimentally investigated. Clean and contaminated drops
took either spherical, spheroidal or deformed spheroidal
shapes when the diameter ratio L was less than a certain
critical value, Lc, whereas they took bullet shapes for L > 2c
(Taylor drops). The applicability of available drag and
Froude number correlations was examined through
comparisons with the measured data. The effects of
surfactants on drop shapes were also discussed. The
conclusions obtained are as follows:

(1) The critical diameter ratio, Lc, strongly depends on the
Edtv6s number Eo, whereas the Morton number, M, and
the viscosity ratio, ug*, do not have much influence on 2c.

(2) The terminal velocities of clean drops are well evaluated
by using the drag correlation of Hayashi & Tomiyama
(2009), a combination of Win Myint's drag correlation
(2006; 2008) and Clift's wall effect multiplier (1978), and


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

the Froude number correlation proposed by Hayashi et al.
(2010b).

(3) The terminal velocities of contaminated drops are well
predicted by taking the limit u<* co in the drag
correlation of Hayashi & Tomiyama (2009) for clean
drops in the viscous force dominant regime.

(4) The reduction in surface tension due to the addition of
surfactants causes the increase in the terminal velocity
and the decrease in the radius of a Taylor drop.

(5) The terminal velocity of a Taylor drop at high EoD and
low * is not affected by surfactants, and therefore, the
Froude number correlation for clean Taylor drops
(Hayashi et al, 2010b) is also applicable to contaminated
drops.

Acknowledgements

This work has been supported by the Japan Society for the
Promotion of Science (grant-in-aid for scientific research (B),
No. 21360084). The authors would like to express our thanks
to Mr. Yasuki Yoshikawa for his assistance in the
experiments and the simulations.

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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

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