Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 17.2.4 - A key parameter to characterize Taylor flow in narrow circular and rectangular channels
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00418
 Material Information
Title: 17.2.4 - A key parameter to characterize Taylor flow in narrow circular and rectangular channels Micro and Nano-Scale Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Wörner, M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: Taylor flow
mini- and microchannels
multiphase capillary reactors
 Notes
Abstract: In this paper we show that the ratio  between the bubble velocity (UB) and the total superficial velocity (Jtot) is a key parameter in Taylor flow. Depending on the value of  the streamlines in the liquid slug show a recirculation pattern or complete bypass flow. Among the quantities that are related to  are the mean liquid velocity, the relative velocity, the gas hold-up, the cross-sectional area of the bypass and recirculation flow region, the non-dimensional recirculation time in the liquid slug, the thickness of the liquid film and the bubble diameter. In experiments and technical applications Jtot is often known or prescribed, whereas UB is unknown. Thus, when  is known, UB and all the other above quantities can be directly computed. By a similitude analysis we show that  may depend on up to ten non-dimensional groups. However, the evaluation of literature data for Taylor flow indicates that it mainly depends on the capillary number. When inertial and gravitational effects are important additionally the Laplace number and the Eötvös number may be of some influence. We thus suggest focusing in future experimental and theoretical studies on further clarification of these functional relationships and propose to correlate  with the capillary number CaJ, which is based on Jtot as velocity scale.
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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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


A key parameter to characterize Taylor flow in narrow circular
and rectangular channels


Martin Wbrner

Karlsruhe Institute of Technology (KIT), Institute of Nuclear and Energy Technologies,
P.O. Box 3640, 76021 Karlsruhe, Germany
martin.w oinmci a kil cdu


Keywords: Taylor flow, mini- and microchannels, multiphase capillary reactors

Abstract

In this paper we show that the ratio V between the bubble velocity (UB) and the total superficial velocity (Jot) is a key
parameter in Taylor flow. Depending on the value of i/ the streamlines in the liquid slug show a recirculation pattern or
complete bypass flow. Among the quantities that are related to y are the mean liquid velocity, the relative velocity, the gas
hold-up, the cross-sectional area of the bypass and recirculation flow region, the non-dimensional recirculation time in the
liquid slug, the thickness of the liquid film and the bubble diameter. In experiments and technical applications Jot is often
known or prescribed, whereas UB is unknown. Thus, when Vy is known, UB and all the other above quantities can be directly
computed. By a similitude analysis we show that y may depend on up to ten non-dimensional groups. However, the
evaluation of literature data for Taylor flow indicates that it mainly depends on the capillary number. When inertial and
gravitational effects are important additionally the Laplace number and the E6tvOs number may be of some influence. We
thus suggest focusing in future experimental and theoretical studies on further clarification of these functional relationships
and propose to correlate V/with the capillary number Caj, which is based on Jot as velocity scale.


Introduction

Taylor flow is a special kind of slug flow in small channels,
where the liquid slugs which separate the elongated
bullet-shaped bubbles (Taylor bubbles) are free from gas
entrainment. Taylor flow occurs in micro-fluidic devices
for applications in life sciences (lab-on-a-chip), material
synthesis and chemical process engineering, e.g. in
catalytic multiphase capillary and monolithic reactors
(Kreutzer et al., 2005). Taylor flow is attractive because of
its well defined interfaces and flow conditions which are
easier to control than in macroscopic devices and because
of its advantageous mass transfer properties. The later
stems from (i) the high interfacial area per unit volume, (ii)
the thin liquid film which separates the body of the bubble
from the channel wall, and (iii) the recirculation in the
liquid slug which accounts for good mixing and a
wall-normal convective transport in laminar low.

In Taylor flow, the liquid film thickness and the
recirculation in the liquid slug depend mainly on the
capillary number CaB = L UB / a, where UB is the bubble
velocity, PL is the liquid viscosity and a is the coefficient
of surface tension. For given gas and liquid superficial
velocities JL and JG the total superficial velocity Jot JL +
JG and the volumetric flow rate ratio p = JG/J are known,
whereas the bubble velocity (and thus CaB) and the
gas-holdup = JG / UB are unknown.

In this contribution we perform a similitude analysis for
incompressible Taylor flow in straight circular and
rectangular channels. We show that many characteristic


quantities in Taylor flow are related to a single key
parameter, namely to the ratio between the (unknown)
bubble velocity and the (given) total superficial velocity.
Thulasidas et al. (1997) denoted this ratio by V/'- U / Jot.
We compare theoretical and experimental results from
literature to elucidate the functional dependence of y' on
the capillary Caj, which is based on Jot as velocity scale.

Problem Description

We consider the pressure-driven flow of two immiscible
fluids with constant physical properties in a straight
channel. The channel cross-section with area Ah is either
circular (diameter D, radius R, area Ah = iD2/4) or
rectangular with width B and height H (area Ah = BH). For
the rectangular channel we assume H ratio =- H/B is in the range 0 < <1. We further define
the hydraulic diameter Dh = 2BH/(B+H).

We assume that the Taylor flow consists of a sequence of
alternating gas bubbles and liquid slugs, where the length
of all gas bubbles and that of all liquid slugs is the same.
Then, the flow hydrodynamics is fully described by a
single unit cell of length Lu, = LB + Lsiug. In Figure 1 we
show a sketch of such a "perfect" Taylor flow.

We denote the constant volumetric flow rates of both
phases by QG and QL. Then, the gas and liquid superficial
velocities are JG -G / Ah and JL -QL / Ah, respectively.
The bubble velocity and the mean liquid velocity are given
by UB J= G/ and UL -JL / (1 e). Here, e = VB / Vu is
the fractional volumetric gas content in the unit cell, and





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


VB and V. = Lu, Ach denote the volumes of the bubble and
the unit cell, respectively. With these definitions, the total
superficial velocity is given by


SAp uc _ch
PL QG +QL


Jtot =Q = +J = B + (1- )U
Ach

For any cross-section at a certain axial position it is

Ach AL y)+ AB(y)


where AL and AB are the cross-sectional areas occupied by
the liquid and the bubble, respectively.

Similitude Analysis

In this section we perform a similitude analysis of the
problem described above and begin with a list of the
relevant quantities for a rectangular channel.

First, there are six (constant) physical properties, namely
* Gas and liquid density (pG, pL)
* Gas and liquid viscosity (pUG, UL)
* Coefficient of surface tension (o)
* Gravitational acceleration (g)

Next, there are three flow specific quantities, namely
* Gas and liquid volumetric flow rate (QG, QL)
* Pressure drop along the unit cell (Ap.u)

Finally, there are five geometrical quantities, namely
* Channel height (H)
* Channel width (B)
* Length of the unit cell (Lo)
* Bubble volume (VB)
* Angle of channel axis with respect to the gravitational
field (cp)

These are in total 14 variables with three basic dimensions,
namely kg, m, s. According to the it-theorem there are
eleven independent non-dimensional groups that
characterize the problem. (For a circular channel it is one
less because the two length scales H and B are replaced by
the single length scale D).

From the above 14 quantities, in an experiment usually the
following eleven quantities are given or prescribed: po, PL,
/G ,PL g ,QG ,QL ,H ,B ,p. The three main unknown
parameters are then the length of the unit cell, the gas
holdup in the unit cell and the pressure drop along the unit
cell. The respective three non-dimensional groups are:


(1) As the eight other independent non-dimensional groups we
choose


H4 =PG
PL


I5 n
/UL


H6 = G /
QGQL



H7 _UL QG+QL
Ach


H8 LDh L
t'L


9g(p pG)D
C9


H
10 B


JULJ, t
a


Thus in an experiment 14 nll are (usually) given, while
the non-dimensional groups H1 13 are unknown.

We note that the non-dimensional unit cell length A
depends in a significant way on the device and mechanism
used to generate the Taylor bubbles. Thus, it is often
possible to adjust an experiment so that A is within a
certain range.

The main global quantities of interest are then the gas
holdup e and the Euler number Euo, which represents the
non-dimensional pressure drop. For both quantities the
following functional relations hold:

E = F,(Euc, A, p', u',f, Ca,, La, Eo, x, o) (14)


Euu, = FE, (E, A, p', p', 6, Ca,, La, Eo, o, o) (15)


L
H1-Lc -UA
Dh


H2 g -
ALuc


The key parameter yV

(4) From the above eleven independent non-dimensional
groups further non-dimensional groups can be defined.
Examples are the Reynolds number


P -Eu
L 2 uc
PLtot





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


Re LDhJtlt LaCa,
t'L


* The bubble Reynolds number


ReB = VRej = VLaCa,


and the Weber number

D 2
We PLDhtt = CaRe = LaCa
a


Here, we have chosen the Laplace number as group H8I
instead of the Reynolds and Weber number because for
given fluid properties and a certain channel hydraulic
diameter La is a constant. The same holds for the Edtv6s
number Eo.

In this paper we suggest as an especially useful
non-dimensional group the ratio


UB P
Jtot s


so that


P
F,
F (Euc, A, p', ,/', ,, Ca La, E6, X, ()


E Eu, (= G A, p', ,/' /, Ca La, Eo, X, (o) (20)

In this paper we will not consider relations for the Euler
number Euo but focus only on the functional relationship
for V/, which is of fundamental importance for Taylor flow.
Namely, when V is known, the following quantities can be
directly computed from other given or prescribed
quantities:


* The non-dimensional relative velocity, both in the
form


SUB -Jtot
UB


1


and in the form

SUB Jtot 1 (27)
Z = = 7-l (27)
Jtot

Next, we show that Vy is also related to important local
quantities in Taylor flow such as the liquid film thickness.

Dependence of bubble diameter and liquid film
thickness on y/

A mass balance for the liquid phase in a frame of reference
moving with the bubble yields

(ot B )Ah = [UL,cs () B ] AL (y) (28)

Here, ULs(y) is the mean cross-sectional liquid velocity at
a certain axial position. In the liquid slug it is AL = Ach SO
that for this case it follows immediately from Eq. (28) the
well known result Us, = Jto, which states that the mean
axial velocity at any cross-section within the liquid slug is
equal to the total superficial velocity. (For the flow of two
incompressible phases through a straight channel with
constant cross-section this result also follows from global
mass continuity, see Suo & Griffith, 1964).

From Eq. (28) it follows with Eq. (2)


* The bubble velocity


UB = VJtt


AL(y) y-1
Ach Ucs (Y)/Jtot


* The mean liquid velocity (this follows from Eq. (1))


AB()_ 1- L,cs (Y)/ott
Ach UL,cs (Y)/tot


UL 1-f
Jtot 1-8/ l

* The gas holdup



SThe capillary number

* The capillary number


If in the liquid film the velocity UL,s(y) is zero, then it
follows from Eqs. (29) and (30) the result


AL AB, 1
L1 B1-- (31)
Ach Ach


Thus, in the case of a stagnant liquid film the
cross-sectional area of the bubble and that of the liquid
film is constant and axially uniform. An example for this
case is the penetration of an inviscid gas phase into a


CaB = VCaj





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


capillary where it displaces a viscous liquid in the absence
of gravity. Indeed, in such a situation the thickness of the
liquid film becomes constant in a distance sufficiently far
from the bubble tip (Bretherton, 1961; Giavedoni & Saita,
1997). In this case the fraction of liquid, m, left behind the
semi-infinite bubble is of interest (Taylor, 1961). Since this
fraction is equal to the ratio AL / Ach it follows from Eq.
(31) and definition (26) that it is m = W in this special case.

For a circular channel where the thickness of the liquid
film, 6, is uniform at the bubble circumferential it is


D2 -(D- 232
AL 4 4


43(D 5F)
D2


4

With this result and Eq. (31) we obtain for the thickness of
the stagnant liquid film the quadratic equation


Fhic hF t 1 so u
D D 4 )

which has the solution


I1 -I
2 yVf


For << D it follows from Eq. (33) the approximation


DB_ 1+Z

Dh J2r


It is important to note that in inclined or vertical channels
and in the presence of gravity, the velocity in the liquid
film UL,csy) will be in general different from zero and will
vary with the axial position. Then, also the liquid film
thickness and the bubble diameter become, according to
Eqs. (29) and (30), a function of y and are no longer
uniform along the body of the bubble. In this case V/ will
depend on the E6tvOs number and on the angle (p.

Recirculation flow and bypass flow

In incompressible Taylor flow in a frame of reference
moving with the bubble a recirculation pattern in the liquid
slug occurs when the bubble velocity is lower than the
liquid velocity on the channel axis, i.e. for UB < ULmax
(Taylor, 1961; Cox, 1964). For a liquid slug with a fully
developed laminar velocity profile the maximum liquid
velocity is given by ULmax = CUsiug, where C is a constant.
The value of C depends on the shape of the channel
cross-section. For a circular channel it is Co = 2 while for a
square channel it is C, = 2.096. Since in incompressible
Taylor flow it is Usiug = Jot, the condition for recirculation
flow becomes UB < CJot, or I/< C, respectively.

The cross-sectional regions with bypass flow (close to the
walls) and recirculation flow (in the channel center) are
separated by the "dividing streamline" (Thulasidas et al.,
1997), see Figure 1. The position of the dividing
streamline is obtained from the condition that the total
flow rate within the recirculation area is zero in the moving
frame of reference. Thulasidas et al. (1997) showed that in
a circular channel the radial position r, of the dividing
streamline is given by


so that


1 43
/= =-- 1+-F
S43 D
1
D


while the radial position ro, where the velocity in the
moving frame of reference is zero, is given by


For the bubble diameter in a circular channel we obtain -
under the assumption of a stagnant liquid film from Eq.
(31) the result


R 2


DB 1
D J


For a rectangular channel and an axisymmetric bubble
surrounded by a stagnant liquid film it follows from Eq.
(31) for the bubble diameter


DB 2
B Ff/ I


DB 2
H 27/'


Thus, in a circular channel the fractional recirculation area
isA1/Ach = 2 yand it isA1/Ao = 2 for any value of y.

The intensity of the recirculation can be quantified by the
time needed for the liquid to move from one end of the
liquid slug to the other end. A second characteristic time
scale is the time needed by the liquid slug to travel a
distance of its own length. Thulasidas et al. (1997) defined
the ratio of both time scales as the non-dimensional
recirculation time, r. For a circular channel the
recirculation time is (Kececi et al. 2009)


or, in terms of the hydraulic diameter


s F I I1 I
D 4 V










1-


For / = Co = 2 it is A1/Ach = 0 and the recirculation area
vanishes. For y> 2 complete bypass flow occurs.

For rectangular channels, the fractional areas A1/Ach and
Ao/Ach as well as the non-dimensional recirculation time r
have been studied theoretically by Kececi et al. (2009).
Based on the assumption that the liquid slug is sufficiently
long to form a fully developed laminar velocity profile, the
authors used an approximation to the exact laminar
velocity profile in a rectangular channel proposed in
literature and showed tlIu. i, ,, Ao/Ach and r all depend
in a unique way on y and on the channel aspect ratio -.
Kececi et al. (2009) evaluated theses relations numerically
and displayed the results in graphical form as function of w/
for different values of -.

The transition to complete bypass flow occurs in a square
channel for y > 2.096 and in a planar channel, formed by
two parallel plates, for y > 1.5. For rectangular channels
the transition to complete bypass flow occurs for y > C_.
The value of C, can be determined from the relation


. =3 (1+0.54669x+1.55201x2
2 (43)
-4.05943 3 +3.2149374 -0.85731,5)

Eq. (43) was proposed by Spiga & Morini (1994) who
determined for laminar single phase flow in a rectangular
channel exact values of Umax / Umean for ten different values
of the aspect ratio in the range 0 < < 1 and fitted these
data by Eq. (43) with an accuracy of 0.06%.

Correlations for Vy

In the previous section we showed that y is related to other
quantities by means of the following identities


UB
J/--
Jtot


7 1
---=+Z
E 1-W


In this section we collect and compare literature data for
different expressions in Eq. (44).

Liu et al. (2005) performed experiments in capillaries with
circular and square cross-section with hydraulic diameters
in the range of 0.9 3 mm using air and three different
liquids in co-current upward flow. They fitted their
experimental data for the ratio of bubble velocity to total
superficial velocity by the correlation


1
S= Ca33
1-0.61C a,


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

Correlations from relative bubble velocity

From experiments in circular tubes, Fairbrother & Stubbs
(1935) suggested the following correlation for m

m = 1.0Ca5 (46)

which is valid in the range 7.5 x 10-5 < CaB < 0.014. Later
Taylor (1961) extended the validity of Eq. (46) for CaB up
to 0.1. With Eq. (26) and with CaB = VCaj Eq. (46)
becomes an implicit relation for w/as function of Caj


-1 = 0CaV (47)


Bretherton's (1961) analytical approach at low CaB
resulted in the following expression

m = 2.68Ca/3 (48)

Giavedoni & Saita (1997) found that their numerical
results for the film thickness match the theoretical
correlation of Bretherton for CaB < 10-3. Thus, we adopt
this range also for the validity of Eq. (48). This equation is
equivalent to


1 =2.68y2/Cat (49)


For large values of CaB, Taylor (1961) found that m
acquires a value of 0.58, while theoretically it was found to
be equal to 0.6 (Cox, 1964). This was confirmed by the
numerical simulations of Giavedoni & Saita (1997) who
found m = 0.559 for CaB = 2 and m = 0.592 for Ca = 10.
The value of m = 0.559 for CaB = 2 corresponds to / =
1/(1-0.559) = 2.268 for Caj = 2 / 2.268 = 0.882 while that
of m = 0.592 for CaB = 10 corresponds to //= 1/(1-0.592)
= 2.451 for Caj = 10 / 2.451 = 4.08.

Correlations from liquid film thickness and bubble
diameter

For a stagnant liquid film Eqs. (34), (36) or (37) can be
used to determine correlations for y from correlations for
9 or DB, respectively. As an example, we mention the
experimental correlations of Marchessault & Mason (1960)
for the thickness of the liquid film in a circular channel
with various inclinations


2- [= AL B 7] (50)
D

where A and B are coefficients (the dimension of A is
cm 5s s5 while B is dimensionless). With Eq. (35) the latter
correlation translates into


This correlation is valid in the range 0.0002 < Caj K 0.39.









1 A
m=l = 20 5Ca05 B
J I UB


where UB is in cm/s. For A = 0 and B = 0.5 Eq. (51)
becomes identical with Eq. (47) from Bretherton (1961).

By scaling arguments for a semi-infinite bubble Aussilous
& Quere (2000) derived the following equation for the
liquid film thickness in a circular tube


0.66CaB
1+3.33C B


which is valid in the range 10-3 < CaB < 1.4. Inserting Eq.
(52) inEq. (34) gives


1


1.32Ca 2/3 1+2Ca213
1I3.33Ca213 1+3.33Ca213


In terms of Caj the latter equation becomes


12 1 2
+3= .+ 332/3 Ca2/ 3
S+ 2l2Ca/3


For square capillaries Kreutzer et al. (2005) proposed the
following correlation for the bubble diameter in the
diagonal direction


DBsq = 0.7 + 0.5exp(-2.25Ca445)
Dh


For CaB > 0.04 the bubble is axisymmetric so that Eq. (39)
can be used to obtain the following implicit equation for Vy



SB (56)

4 07+0.5exp (2.25u 45Ca 445


In Eqs. (45), (47), (49), (54) and (56) taken from literature,
the velocity ratio wis correlated only with 17 = Caj but not
with any other non-dimensional group. In Figure 2 we
display the functional relations y = Vy (Caj) as obtained by
Eqs. (45), (47), (49), (54) and (56). Also shown are
experimental data of Thulasidas et al. (1995) for co-current
upward Taylor flow in a square channel (Dh = 2 mm).
These data show a large scatter band for Caj < 0.01 while
for higher values of Caj the scatter of the data is reduced
and with increase of Caj an increase of Vy can be observed.
For Caj < 0.03 all correlations which are valid in this range
give about similar values for Vy and the deviations are in
general small. Also the value of Vy is less than 1.2 in this
range, indicating that the bubble moves only slightly faster
than the total superficial velocity. For Caj > 0.03 a rather


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

good agreement is found between the correlations of
Aussilous & Quere (2000), Eq. (54), and Kreutzer et al.
(2005), Eq. (56), and the experimental data of Thulasidas
et al. (1995), whereas the correlation proposed by Liu et al.
(2005) for their experimental data, Eq. (45), yields much
smaller values. The correlation of Kreutzer et al. (2005)
extends to values up to Caj = 3.86 and approaches a
limiting value of y= 2.59. So it seems a reasonable good
fit for square channels for Caj > 0.03 when the bubble is
axisymmetric. A comparison of the correlations of
Aussilous & Quere (2000) for a circular channel, Eq. (54),
and the one of Kreutzer et al. (2005) for a square channel,
Eq. (56), suggests that for a given value of Caj the value of
V/ is slightly larger in the square than in the circular
channel.

Conclusions

In this paper we showed that the ratio between the bubble
velocity (UB) and the total superficial velocity (Jtot), which
is here denoted as y, is a key parameter in Taylor flow.
Among the quantities that are uniquely related to V/ are the
mean liquid velocity, the relative velocity, and the gas
hold-up. Depending on the value of V, the flow in the
liquid slug shows in a moving frame of reference a
recirculation pattern in the channel center and bypass flow
close to the walls or complete bypass flow. The
cross-sectional area of the bypass and recirculation region,
and the non-dimensional recirculation time in the liquid
slug depend on Vy and, for rectangular channels, on the
channel aspect ratio. The local thickness of the liquid film
in a certain cross-section depends on the mean liquid
velocity in this cross-section and on y. If the liquid film is
stagnant, unique relations exists between the liquid film
thickness and bubble diameter and i/. A similitude analysis
showed that Vy may depend on up to ten other
non-dimensional groups. However, the evaluation of
literature data for Taylor flow indicates that V/ mainly
depends on the capillary number. While these data show
some scatter, they nevertheless give a consistent picture of
this dependence.

In experiments and technical applications the total
superficial velocity is often known or prescribed, whereas
the bubble velocity is unknown. Thus, when Vy is known,
all the above quantities can be directly computed. When
inertial and gravitational effects are important, Vy will in
addition to the capillary number also depend on the
Laplace number (La) and the E6tv6s number (Eo). Also the
volumetric flow rate ratio (f) might be of some influence.
We thus propose to focus in future experimental and
theoretical studies on further clarification of the functional
relationship Vy = Vy (Caj, ,8, La, EO). These four parameters
are easy to vary and control in experiments. Furthermore,
V/ can be determined from the bubble velocity, which is
easy to measure in transparent channels.





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


Ao cross-sectional area where the velocity in the
liquid slug is positive in a frame of reference
moving with the bubble (m2)
A1 cross-sectional area of recirculation region (m2)
AB cross-sectional area of bubble (m2)
Ah channel cross-sectional area (m2)
AL cross-sectional area occupied by liquid (m2)
B width of rectangular channel (m)
Ca Capillary number (-)
D diameter of circular channel (m)
Dh hydraulic diameter of rectangular channel (m)
Eo E6tv6s number (-)
Euc, Euler number (-)
g gravitational constant (in S:)
H height of rectangular channel (m)
JG gas superficial velocity (m/s)
JL liquid superficial velocity (m/s)
Jto total superficial velocity (m/s)
La Laplace number (-)
LB bubble length (m)
Lslug length of the liquid slug (m)
L, Length of the unit cell (m)
Apo, pressure difference along the unit cell (Pa)
QG gas volumetric flow rate (m3/s)
QL liquid volumetric flow rate (m3/s)
R radius of circular channel (m)
Re Reynolds number (-)
UB bubble velocity (m/s)
UL mean liquid velocity within the unit cell (m/s)
siug mean velocity in liquid slug (m/s)
V volume (m3)
W non-dimensional relative velocity (-)
We Weber number (-)
Z non-dimensional relative velocity (-)

Greek letters
X aspect ratio of rectangular channel (-)
$ liquid film thickness (m)
e gas holdup in unit cell (-)
(p angle between channel axis and gravity vector
(-)
A non-dimensional length of the unit cell (-)
Lp dynamic viscosity (Pa s)
p density (kg/m3)
o- coefficient of surface tension (N/m)
z non-dimensional recirculation time (-)
y/ velocity ratio /- UB/ Jtot (-)

Subscripts
B bubble
F liquid film
G gas phase
J quantity is based on Jo as velocity scale
L liquid phase
slug liquid slug
uc unit cell


Aussilous, P., Quere, D. Quick deposition of a fluid on the
wall of a tube. Phys. Fluids 12 (2000) 2367.

Bretherton, F.P. The motion of long bubbles in tubes. J.
Fluid Mech. 10 (1961) 166.

Cox, B.G. An experimental investigation of the streamlines
in viscous fluid expelled from a tube. J. Fluid Mech. 20
(1964) 193.

Fairbrother, F., Stubbs, A.E. The bubble-tube method of
measurement. J. Chem. Soc. 1 (1935) 527.

Giavedoni, M.D., Saita, F.A. The axisymmetric and plane
cases of a gas phase steadily displacing a Newtonian liquid
- A simultaneous solution of the governing equations. Phys.
Fluids 9 (1997) 2420.

Kececi, S., Wrner, M., Onea, A., Soyhan, H.S.
Recirculation time and liquid slug mass transfer in
co-current upward and downward Taylor flow. Catalysis
Today 147S (2009) S125.

Kreutzer, M.T., Kapteijn, F., Moulijn, J.A., Heiszwolf, J.J.
Multiphase monolith reactors: chemical reaction
engineering of segmented flow in microchannels. Chem.
Eng. Sci. 60 (2005) 5895.

Liu, H., Vandu, C.O., Krishna, R. Hydrodynamics of
Taylor flow in vertical capillaries: flow regimes, bubble
rise velocity, liquid slug length, and pressure drop. Ind.
Eng. Chem. Res. 44 (2005) 4884.

Marchessault, R.N., Mason, S.G. Flow of entrapped
bubbles through a capillary. Ind. Engng. Chem. 52 (1960)
79.

Spiga, M., Morini, GL. A symmetric solution for velocity
profile in laminar flow through rectangular ducts. Int.
Commun. Heat Mass Transfer 21 (1994) 469.

Suo, M., Griffith, P. Two-phase flow in capillary tubes.
ASME J Basic Eng. 86 (1964) 576.

Taylor, G.I. Deposition of a viscous fluid on the wall of a
tube. J. Fluid. Mech. 10 (1961) 161.

Thulasidas, T.C., Abraham, M.A., Cerro, R.L. Bubble-train
flow in capillaries of circular and square cross section.
Chem. Eng. Sci. 50 (1995) 183.

Thulasidas, T.C., Abraham, M.A., Cerro, R.L. Flow
patterns in liquid slugs during bubble-train flow inside
capillaries. Chem. Eng. Sci. 52 (1997) 2947.


Nomenclature


References




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


A,(y) S,(y) Ach -


region with


Figure 1: Sketch of Taylor flow with characteristic dimensions, areas, and streamlines in the recirculation flow regime
(recirculation flow pattern after Taylor (1961)).


+ Thulasidas etal. (1995)
Eq. (45) --- Eq. (47)
Eq. (49) -- Eq. (54)
------ Eq.(56) G.&S. (19!



square channel:
non-axisymmetric bubble



++ + + 4A
++ +

+ tsquar


37)


S,4- bypass
t bypass


recirculation flow |


re channel:


axisymmetric bubble


1 E-4


1E-3 0.01 0.1 1


I 10
10


Ca, [-]
Figure 2: Literature correlations for V/ as function of the capillary number Caj. The symbol u corresponds to the numerical
data of Giavedoni & Saita (1997) for CaB = 2 and 10, respectively. The thin grey horizontal lines indicate the transition
from recirculation flow to complete bypass flow in a circular (lower line) and a square (upper line) channel. The thin grey
vertical line indicates the transition from a non-axisymmetric to an axisymmetric bubble shape in a square channel.


A,(y)


3.0



2.5



2.0


1.5



1.0


flow


I




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