4884 Ind. Eng. Chem. Res.

2005, 44, 4884-4897

Hydrodynamics of Taylor Flow in Vertical Capillaries: Flow

Regimes, Bubble Rise Velocity, Liquid Slug Length, and Pressure
Drop
Hui Liu,† Chippla O. Vandu, and Rajamani Krishna*
Van’t Hoff Institute for Molecular Sciences, University of Amsterdam, Nieuwe Achtergracht 166,
1018 WV Amsterdam, The Netherlands

Two-phase flow hydrodynamics in vertical capillaries of circular and square cross sections were
experimentally studied, using air as the gas phase and water, ethanol, or an oil mixture as the
liquid phase. The capillary hydraulic diameters ranged from 0.9 mm to 3 mm, with the superficial
gas and liquid velocities covering a span of 0.008-1 m/s, which is typical of that obtained in
monolith reactors. Using a high-speed video camera, four distinct flow regimes were observed
within the range at which experiments were conducted: bubbly, slug-bubbly, Taylor, and churn
flows. Annular flow was observed at excessively high gas and low liquid flow rates, well beyond
those of interest to this study. Based on the definition of a two-class flow regime, the combination
of two parameterssthe slip ratio (S) and the ratio of the superficial gas velocity to two-phase
superficial velocity (UG/UTP)swas observed to be suitable for determining the transition from
homogeneous flow to nonhomogeneous flow. The influence of capillary geometry, capillary
hydraulic diameter, and fluid properties on bubble rise velocity (Vb) were investigated and
determined to be of little significance. Furthermore, a new and simplified correlation for
predicting Vb and, by implication, the gas holdup (G) was proposed. Liquid slug lengths were
also experimentally studied, using a correlation that was developed to estimate them. Pressure
drop experiments were also performed, and the peculiar phenomenon of negative frictional
pressure drops was observed at very low liquid velocities. By defining a new dimensionless
quantitysthe pressure factor, FEsa flow-regime-dependent method for estimating the total
pressure drop in two-phase vertical capillary flows was developed.

1. Introduction provided that a uniform gas and liquid distribution

(such as that obtained) occurs in the monolith bundle.16
Monolith reactors are becoming increasingly signifi- Certain multiphase flow characteristics in capillaries
cant as multiphase reactors, considering the advantages have been studied by several investigators, such as gas-
that they offer, in comparison to conventionally used liquid flow hydrodynamics,2,17-20 mass transfer,2,21-24
trickle beds and slurry bubble columns for a host of and reaction rates.10,25 These studies were generally
processes.1-5 These advantages, which include low- performed in the Taylor flow regime, which has been
pressure drop, high gas-liquid mass-transfer rates, and reported to be an effective regime for the operation of
minimum axial dispersion, stem from the uniquely
monolith reactors. Taylor flow, which is also known as
structured multichannel configuration of monoliths.
slug flow or bubble train flow, is characterized by the
Some studies have shown that the utilization of mono-
presence of elongated gas bubbles with lengths greater
lith reactors, in lieu of trickle beds, results in higher
than the capillary diameter, which rise along the
productivities and a very significant reduction in reactor
capillary separated from each other by liquid slugs. The
size for specified processes.6,7 Several other studies have
gas bubbles occupy most of capillary cross section,
focused on the use of monolith reactors for hydrogena-
separated from the channel wall by a thin liquid film.
tion,8-10 hydrodesulfurization,11 and oxidation12,13 reac-
tions, as well as the Fischer-Tropsch synthesis.14,15 This flow arrangement has been reported to yield
superior mass-transfer performance.17
In essence, a monolith block is composed of an array
of uniformly structured parallel channels, often of Within the past decade, several two-phase capillary
square or circular geometry, typically having hydraulic studies have focused on one or more of flow regimes,
diameters between 1 mm and 5 mm. Thus, the monolith bubble velocity, and pressure drop. Many of these
can be viewed as a structure that is comprised of many investigations, which involved the use of horizontal
repeating building blocks, where the basic building block capillaries, were limited to air-water systems26-29 and
is a single channel. It can be argued that data obtained were geared toward a better understanding of two-phase
from studies on a single channel, or what may be called flow parameters for practical applications in compact
a capillary, can be used in scaling up a monolith reactor, heat exchangers, as well as refrigeration and air con-
ditioning systems. Other flow regime, bubble velocity,
* Author to whom correspondence should be addressed.
and pressure drop investigations have involved the use
Fax: + 31 20 5255604. E-mail: R.Krishna@uva.nl. of vertically positioned capillaries.30-32 Few investiga-
†
Permanent address: College of Chemical Engineering, tors have used liquids other than water.2,17,19,33 Taking
Beijing University of Chemical Technology, Beijing 100029, stock of the various studies that have been undertaken,
China. one observes that limited data are available for capil-
10.1021/ie049307n CCC: $30.25 © 2005 American Chemical Society
Published on Web 12/16/2004
 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 4885

Figure 2. Details of the tee connection.

Table 1. Experimental Systems Studied

gas-liquid capillary hydraulic number of
system type diameter (mm) experiments
air-water circular 0.91 11
air-water circular 2.00 24
Figure 1. Schematic representation of the experimental setup. air-water circular 3.02 44
air-water square 2.89 39
laries of varying geometry and for fluids with surface air-ethanol circular 0.91 17
tension and viscosity values that differ from those of air-ethanol circular 2.00 25
water. The objective of the present investigation is the air-ethanol circular 3.02 17
air-ethanol square 0.99 27
systematic study of the hydrodynamics of gas-liquid air-ethanol square 2.89 41
flow in vertical capillaries of circular and square cross
sections and the development of correlations for the air-oil mixture circular 3.02 37
air-oil mixture square 2.89 24
prediction of related flow parameters such as bubble rise
velocity, liquid slug length, and pressure drop. Air was Table 2. Physical Properties of Liquids Used at 298 K
used as the gas phase, with water, ethanol, or an oil and 100 kPa
mixture being used as the liquid phase. The gas and liquid density viscosity surface tension
liquid superficial velocities were varied in the range of phase (kg/m3) (mPa s) (mN/m)
0.008-1 m/s. Within this range, a broad spectrum of
water 998 0.95 72
flow regimes was encountered. However, particular ethanol 780 1.2 22
interest would be focused on the Taylor flow regime, in oil mixture 840 15.9 28
view of understanding its significance to monolith
reactor applications. Liquid densities were determined using a density meter
(PAAR model DMA 35, Austria). Liquid viscosities and
2. Experimental Section surface tensions were measured with a Ubbelohde
viscosity meter and by the capillary method, respec-
A schematic representation of the experimental setup tively. During experiments, compressed air was fed
is depicted in Figure 1. It consists of (1) a capillary through a precalibrated float-type gas flowmeter to the
setup, (2) an image recording and analysis system, and tee connection. Two manually operated control valves
(3) a pressure drop measurement system. The capillary were used to regulate the gas flow rate. The first of these
setup is composed of a vertically mounted, 1.4-m-long, valves was placed between the compressed air flow line
single Pyrex glass capillary. Gas and liquid were fed to and the flowmeter and was set to give a constant gauge
the bottom of the capillary through a 3-mm-diameter pressure of 30 kPa. The second valve, which was placed
polyvinyl chloride (PVC) tee connection, as shown in between the gas flowmeter and the tee connection, was
Figure 2. Five different capillaries were used in the solely used to regulate the volumetric gas flow rate into
experiments, and the capillary dimensions and an the capillary. Liquid was fed from an elevated 10-L
overview of the experimental systems studied is shown storage vessel into a precalibrated liquid flowmeter with
in Table 1. Air was used as the gas phase in all the flow rate also adjusted through the use of a
experiments, with the liquid phase being demineralized manually operated valve. Gravity provided the driving
water, ethanol, or an oil mixture that was a miscible force for the flow of liquid from the storage vessel into
blend of a light paraffin oil (viscosity of µL ) 2.9 mPa s, the liquid flowmeter. The liquid flowmeter was a float-
surface tension of σ ) 28 mN/m) and a very viscous type flowmeter that had been installed upstream of the
hydrocarbon oil (µL ) 75 mPa s, σ ) 28 mN/m) in a tee connection. The gas-liquid flow stream arrange-
volume ratio of 1:2. All experiments were conducted at ment allowed for an independent alteration of the gas
room temperature and atmospheric pressure. Physical and liquid flow rates. Liquid was discharged from the
properties of the liquids used are given in Table 2. capillary into a 33-mm-wide, 40-mm-high disengage-
4886 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

ment zone. The height of liquid in the disengagement to Figure 1). For each liquid used, a calibration curve
zone was ∼8 mm. showing the linear relationship between pressure dif-
The image recoding system consists of a Photron ference and the voltmeter readings was made. When the
model Fastcam-ultima 40K high-speed video camera, a capillary was full of a stationary liquid phase, the
memory box, and a CRT monitor display. The high- pressure difference was zero (provided that the same
speed camera system was used for flow-regime observa- liquid as that used in the capillary also fills the tubes
tions and bubble rise velocity measurements. Video that are connected to the pressure transducer) with a
movies captured by the high-speed camera were instan- zero voltage reading displayed on the voltmeter as well
taneously stored in the memory box. The camera could as being fed to the PC. When only air was present in
be set to capture movies at rates of 30-4500 frames per the capillary, however, pressure difference became equal
second (fps) in full-frame mode and 9000-40 500 fps in to the 1.4-m height of the liquid in the tubes that were
segmented-frame mode. The CRT display showed, in connected to the pressure transducer. In this case, the
real time, what was viewed through the high-speed span of the voltmeter was set to 10 V. During experi-
camera. Data from the memory box were transferred ments, pressure drop measurements were performed for
to the personal computer (PC) for later analysis. During a couple of minutes at a sampling frequency of 10 Hz.
the experiments, the high-speed camera was positioned The experimental system was tested by comparing
midway along the capillary height with its focus ad- experimental liquid-only frictional pressure drop data
justed in such a way that it captured rising air bubbles with theoretical frictional pressure drops for laminar
and liquid slugs within a distance of 0.035-0.2 m, flow. For single-phase laminar flow in a vertical capil-
depending on the particular operating gas and liquid lary, the total pressure drop (∆PT) is composed of two
velocities. After steady state was achieved, movies were contributions: (1) the pressure drop due to frictional
made for a certain time span, which varied with bubble effects of the liquid flow (∆Pf) and (2) the hydrostatic
velocity, at capture rates of 250-4500 fps, depending pressure drop of the liquid. Experimental ∆Pf values
on the gas and liquid velocities, to obtain suitable movie were obtained by subtracting the hydrostatic con-
time intervals. By performing a frame-by-frame analysis tribution from the measured total pressure drop. The
of each movie, the bubble frequency (fb), which is defined theoretical ∆Pf value was computed, noting that, for
as the number of bubbles that traverse a given point in laminar flow, the Fanning friction factor (fL) is related
the capillary per unit time, was determined. The bubble to the liquid-phase Reynolds number (ReL) by the
rise velocity (Vb) was also determined from the movies relation
by registering the time required for a gas bubble to rise
a known distance along the capillary height. Each Vb C
value reported is an average of three to five values that fL ) (4)
ReL
were taken. Nevertheless, hardly any of the values
differed from the mean value by >4%, indicating that
where C is a constant that is dependent on channel
the flow was steady. With the bubble rise velocity
geometry and has values of 14.2 and 16 for square and
known, the gas holdup (G) was determined as follows:
circular channels, respectively. The Fanning friction
UG factor is related to the frictional pressure drop by the
G ) (1) relation
Vb
∆Pf/Lc
where UG is the superficial gas velocity. fL ) 1
(5)
In the Taylor flow regime of a vertical upflow capil- /2FLUL2(4/dc)
lary, a gas bubble rises along the capillary height
sandwiched between liquid slugs. Between the gas where dc and Lc are the diameter and length of the
bubble and the wall of the capillary is a thin downflow- capillary, UL is the superficial liquid velocity, and FL is
ing liquid film. We define a unit cell as consisting of a the liquid density. Figure 3 shows the experimental
gas bubble and the accompanying liquid slug beneath single-phase frictional pressure drop with predictions
it. Furthermore, we assume that, on average, all unit obtained using the fL values for laminar flow. A very
cells have the same length. The average unit cell length good agreement is obtained, verifying the accuracy of
(LUC) can then be estimated from the setup and also helping to ascertain that inlet and
outlet effects are negligible.
Vb All the measured experimental data for 11 different
LUC ) (2) campaigns with varying system properties and capillary
fb
sizes, and configuration, have been tabulated and
Assuming that the volume of liquid in the film between presented in the Supporting Information accompanying
the gas bubble and capillary wall is negligible, the this publication.
average liquid slug length (Lslug) can be estimated from
the relationship 3. Results and Discussion

Lslug ) LUC(1 - G) (3) 3.1. Two-Phase Flow Regimes. Typical images of
flow regimes observed during experiments are shown
The pressure drop in the capillary was measured in Figure 4 for air-water, air-ethanol, and air-oil
using a differential pressure transducer. Two liquid mixture systems in the 3.02-mm-diameter circular
taps, at the base and top of the capillary, were connected capillary. A total of five distinct flow patterns were
to pressure ports on a Validyne model DP15 pressure observed and are labeled bubbly flow, Taylor flow, slug-
transducer, which was, in turn, connected to an analog- bubbly flow, churn flow, and annular flow. A basic
to-digital converter card on a PC via a voltmeter (refer description of each flow regime follows.
 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 4887

Figure 3. Experimental and theoretical single-phase frictional pressure drop for the (a) 2.89-mm square capillary (water), (b) 2-mm
circular capillary (ethanol), and (c) 0.99-mm square capillary (ethanol).

in the liquid slugs; this is a feature that is not observed

in Taylor flow. The transition from Taylor flow to slug-
bubbly flow occurs by increasing the liquid flow rate
with the gas flow rate being kept constant.
Churn Flow Regime: Churn flow occurs at very high
gas velocities. It consists of very long gas bubbles and
relatively small liquid slugs. Because of the high gas
velocity, a wave or ripple motion is often observed at
the bubble tail with tiny gas bubbles entrained in the
liquid slug. Further increases in the gas flow rate result
in annular flow.
Annular Flow Regime: At excessively high gas veloci-
ties and very low liquid velocities, annular flow results.
Here, a continuous gas phase is present in the central
core of the capillary with the liquid phase being dis-
placed to form an annulus between the capillary wall
and the gas phase.
These flow regimes are what may be called “clear-
cut” regimes, because they are easily identifiable. Other
investigators have also observed somewhat similar
patterns in vertical and horizontal capillaries.26,30,32,34
A degree of complexity becomes involved when identify-
ing flow patterns that fail to fall into any of the typical
regimes depicted in Figure 4. Such patterns can be
expected to occur when transitioning from one regime
Figure 4. Flow regimes encountered during the experiments to another, and, in such cases, the discretion of the
performed for air-water, air-ethanol, and air-oil mixtures. investigator is heavily relied upon. The slug-bubbly flow
Regimes shown are (i) bubbly, (ii) slug-bubbly, (iii) Taylor, (iv) regime was not observed for the viscous oil mixture used
churn, and (v) annular flow. High-speed video recordings of the
various flow regimes can be viewed on our website.35
in this study. Annular flow was not encountered within
the range of gas and liquid superficial velocities for
Bubbly Flow Regime: This flow pattern typically which experimental data are presented. Also, all ex-
occurs at relatively high liquid velocities and low gas perimental liquid slug lengths reported in this work
velocities. It is characterized by the presence of fast are for the Taylor flow regime. Movies of the flow
rising bubbles with diameters less than or equal to the regimes depicted in Figure 4 can be viewed on our
capillary diameter. The bubbles are often spherical or website.35
spheroidal in shape. Gas holdups are generally very low The flow regime map for the 3.02-mm-diameter
in this regime. circular capillary air-water system obtained from the
Taylor Flow Regime: Also known as slug flow or experiments that have been performed is shown in
bubble train flow, Taylor flow consists of gas bubbles Figure 5, with the superficial gas and superficial liquid
with lengths greater than the tube diameter that move velocities as the x- and y-axes. Furthermore, compari-
along the capillary separated from each other by liquid sons are made with some flow regime maps available
slugs. Depending on the gas and liquid flow rates and in the literature. Zhao and Bi32 studied flow regimes in
properties, the bubbles often have hemispherically vertical triangular capillaries for co-current upward
shaped tops and flattened tails. air-water two-phase flow. Their experimental data for
Slug-Bubbly Flow Regime: This is a transition regime a 2.886-mm hydraulic diameter capillary is presented
that occurs between bubbly and Taylor flows. Similar for comparison of the bubbly-slug and slug-churn tran-
to that observed in Taylor flow, bubble slugs are present, sitions. As can be seen, the slug-churn transition occurs
separated from each other by liquid slugs. However, in at very close superficial gas and liquid velocities in both
the slug-bubbly regime, small bubbles are also present capillaries. However, a significant difference exists in
4888 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

Figure 5. Flow-regime map for the 3.02-mm circular capillary air-water system.

the bubbly-slug transition, a discrepancy that could ous liquid phase, which, provided that the liquid velocity
largely be due to the different gas-liquid distri- is not too high, maintains a Hagen-Poiseuille flow
butors used. Although a tee gas-liquid inlet was used pattern. On the other hand, Taylor, churn, and annular
in this study, they used a fine-plastic-packed porous flows are generally nonhomogeneous, with slug-bubbly
mixer to ensure that gas and liquid were well-mixed flow bordering the homogeneous and nonhomogeneous
before entering the capillary. Correlations provided by regimes. From eq 6, we note that, if S ≈ 1, homogeneous
Mishima and Ishii36 for predicting flow-regime transi- flow is obtained, because, under this condition, Vb ≈ VL,
tions for upward two-phase flow in vertical capillaries which is a situation that can only occur when gas and
are also compared with our experimental data in Figure liquid rise uniformly in the capillary with no down flow
5. Neither the bubbly-slug transition nor slug-churn of the liquid phase. When S > 1, bubbles rise with
transition correlations predict our experimental data higher velocities than the liquid phase. This can be
satisfactorily, with the former showing much larger expected when down flow of the liquid phase occurs in
deviation. Interestingly, they also found that, although the liquid film surrounding the bubbles, such as that
these correlations predicted the experimental data of a observed in the Taylor and churn flow regimes. In a
few investigators, they were not suitable at all for nutshell, S can serve as an indication of when a two-
predicting the experimental data of other investigators, phase flow deviates from the homogeneous flow regime.
partially attributing this to different methods of obser- To buttress this point, Figure 6a was re-plotted using
vations and definitions of flow regimes as well as the UG/UTP and G as coordinates in Figure 6b and UG/UTP
fact that transition phenomena develop gradually. The and S as coordinates in Figure 6c. UTP is the two-phase
bubbly-slug transition line, based on the correlation of superficial velocity, which is defined as
Suo37 from studies in a horizontally positioned capillary
for air-water flow, is also depicted in Figure 5. It UTP ) UG + UL (7)
qualitatively predicts the trend for transition from the
bubbly to slug (Taylor) flow regime. From these plots, it is seen that, for UL ) 0.012 m/s
3.2. Gas Holdup and Bubble Rise Velocity. 3.2.1. and UL ) 0.043 m/s, both of which belong to the Taylor
Variation of Gas Holdup and Bubble Rise Velocity. flow regime, deviation from homogeneous flow is most
The variation of gas holdup G with superficial gas significant, judging by the great increase in S. At higher
velocity UG for varying superficial liquid velocities UL liquid velocities, the flows gradually approach homoge-
is shown in Figure 6a for the 2.89-mm square-capillary neous flow, with S values approaching unity. Another
air-ethanol system. An increase in G results from an characteristic of the large deviation from homogeneous
increase in UG and a decrease in UL. At lower values of flow is that it occurs at high gas holdups (G > 0.5), as
UL, local maximums in G are observed. This behavior observed from a comparison of Figure 6b and 6c. For G
is often observed in bubble columns and is generally ) 0.1-0.5, the mean S value is ∼1.4. Also, the local
associated with a flow regime transition. A parameter maximums in G shown in Figure 6a can be attributed
that is often used in two-phase flow analysisscalled the to the significant increase in slip between phases. In
slip ratio, Sshelps to provide some insight into the Figure 6d, the flow regime map for the system is shown;
relationship between the observed trends in G and flow lines that correspond to UG/UTP values of 0.2, 0.5, and
regime transition. The slip ratio S, which is defined as 0.8 also are depicted. From this map, it is clear that
the increase in S occurs near the churn flow regime and
Vb UG/G in the Taylor flow regime. The high S operation is
S) ) (6) predominantly in the Taylor flow regime, whereas a
VL UL/(1 - G) constant S value is observed for the bubbly and slug-
bubbly flow regimes.
is a measure of the relative velocity between the gas Plots of the two-phase superficial velocity versus the
and liquid phases (where VL is the liquid-phase velocity). bubble rise velocity are shown in Figure 7, to demon-
At this point, we can speak of two main classes of flow: strate the effect of certain operating parameters and
homogeneous and nonhomogeneous. Based on the five conditions on the bubble rise velocity. Figure 7a shows
different flow regimes previously discussed, it can be that a linear dependence exists between UTP and Vb.
expected that bubbly flows are typically homogeneous, Moreover, although Vb shows a strong dependence on
because discrete gas bubbles are entrained in a continu- UTP (i.e., the combined gas and liquid superficial veloci-
 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 4889

Figure 6. (a) Variation of gas holdup (G) with superficial gas velocity (UG) and superficial liquid velocity (UL) in the 2.89-mm square
capillary air-ethanol system. (b) UG/UTP versus G for the 2.89-mm square capillary air-ethanol system at different liquid velocities. (c)
Variation of the slip ratio (S) with UG/UTP for the 2.89-mm square capillary air-ethanol system at different liquid velocities. (d) Flow-
regime map for the 2.89-mm square capillary air-ethanol system.

ties), it shows no dependence on the individual gas and noticeable for UTP > 0.6 m/s, where the higher surface
liquid superficial velocities. The effect of capillary tension liquid results in a lower bubble rise velocity,
geometry on bubble rise velocity is shown in Figure 7b, although this effect seems to be only slightly pro-
using air-water data for the 2.89-mm square and 3.02- nounced.
mm circular capillaries. All data points seem to collapse 3.2.2. Correlating the Taylor Bubble Rise Veloc-
to a single line, indicating the negligible effect of ity. Many approaches have been proposed in the
capillary geometry on bubble rise velocity. Data from literature for estimating bubble rise velocities in a
the 0.91-mm and 3.02-mm circular capillaries for the capillary tube. One of these, which is similar to the drift
air-ethanol system are plotted in Figure 7c, to under- flux model used for studies in larger channels, involves
stand the effect of capillary scale on bubble rise velocity. relating the bubble rise velocity to the two-phase
From this plot, bubbles seem to rise more slowly in the superficial velocity:30,38
smaller-diameter capillary. This is likely due to an
increase in surface tension effect with a decrease in Vb ) C1UTPC2 (8)
capillary size. In Figure 7d, the effect of liquid viscosity
on bubble rise velocity is explored using data from the where C1 and C2 are arbitrary constants. However,
3.02-mm circular capillary for air-ethanol and air-oil these arbitrary constants are dependent on such pa-
mixture systems. The oil mixture has a viscosity that rameters as the tube diameter and liquid proper-
is 13 times greater than that of ethanol, whereas the ties, serving as a drawback to this method. Another
two liquids have similar surface tension values (refer method for estimating the bubble rise velocity involves
to Table 2). The higher the liquid viscosity, the higher conducting a mass balance about a unit cell for a fully
the bubble rise velocity for the same UTP. A similar developed steady Taylor flow, resulting in the relation-
result was reported by Kreutzer2 for a 2.3-mm-diameter ship
circular capillary with air and tetradecane as the gas
and liquid phases. Tetradecane has a viscosity that is Lf(Vb - Vf) ) Vb - UTP (9)
3 times greater than that of water. The effect of surface
tension is depicted in Figure 7e for air-ethanol and air- where Lf and Vf are the liquid film holdup and liquid
water systems in the 2-mm-diameter circular capillary. film velocity, respectively. Based on the works of
Ethanol has a surface tension that is approximately a Thulasidas et al.17 and Barnea,39 independent estima-
third of that for water, whereas both liquids have tions of Lf and Vf can be made from which the bubble
approximately the same viscosity (refer to Table 2). This rise velocity can, in turn, be determined. However,
plot shows that the surface tension effect only seems this approach is cumbersome, because it requires that
4890 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

Figure 7. (a) Two-phase superficial velocity (UTP) versus bubble rise velocity (Vb) for the 2.89-mm square capillary air-ethanol system.
(b) Influence of capillary geometry on Vb, using air-water data. (c) Influence of capillary scale on Vb, using air-ethanol data. (d) Influence
of liquid viscosity on Vb in the 3.02-mm circular capillary. (e) Influence of surface tension on Vb in the 2-mm circular capillary.

the bubble diameter, which is a very difficult hydro- based on the experimental work of Heiszwolf et al.40 in
dynamic parameter to measure, be estimated. Further a 200 cpsi monolith reactor, is
discussion on the application of this method for estimat-
ing the bubble rise velocity and comparison with the L
experimental results of this study is available in Ap- Ψslug ) (11)
pendix II of the Supporting Information accompanying
-0.00141 - 1.556L2 ln(L)
this paper.
To overcome difficulties associated with estimating where Ψslug is the dimensionless liquid slug length (Ψslug
the bubble diameter, as well as accounting for the effect ) Lslug/dc) and L is the liquid holdup. Laborie et al.19
of liquid properties on bubble rise velocity, the following studied gas-liquid flow in vertical capillaries and
practical relationship was derived from the correlation correlated liquid slug length data using the following
of all our experimental data: formula:

Vb
)
1
UTP 1 - 0.61Ca0.33
(10)
Ψslug ) 3451 ( 1
Re′GEö )
1.2688
(12)

where Re′G is a gas-phase Reynolds number and Eö is

Ca is the capillary number, which is defined as µLUTP/ the Eötvös number. From observations made during
σ. Figure 8 shows plots of experimental bubble rise this study, as well as information available in the
velocities, as well as predictions obtained using eq 10 literature,2 the liquid slug length would also seem to
for capillaries of circular and square geometries and be influenced by the gas-liquid feed system used. That
different liquids. The fits seem to be very good. The is to say, for a given superficial gas and liquid velocity,
correlation given by eq 10 is valid for predicting Vb in a variation in the configuration of the gas-liquid nozzle
Ca range of 0.0002-0.39. results in a change in liquid slug length on the same
3.3. Liquid Slug Length. The liquid slug length is setup. Such complexity makes the development of a
an important hydrodynamic parameter that has been generally applicable theoretical model for the predic-
reported to have a very significant effect on gas-liquid tion of the liquid slug length difficult. However, a
mass transfer in capillaries.21 It has a very complicated correlation would be developed based on the Taylor-flow-
relationship to system parameters such as the super- regime experimental data obtained in this study and
ficial gas and liquid velocities and fluid properties. To comparisons made with the literature correlations of eqs
date, only a few experimental correlations for evaluating 11 and 12. As a first step in doing this, reference is made
liquid slug lengths in Taylor flow are available in the to the vertical capillary mass-transfer study of Bercic
literature. One of these, as suggested by Kreutzer2 and et al.,21 in which the gas-liquid mass-transfer
 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 4891

Figure 8. Experimental and predicted bubble rise velocity (Vb) values for different capillary geometries and liquids used: (a) 2-mm
circular capillary, air-water; (b) 3.02-mm circular capillary, air-water; (c) 2.89-mm square capillary, air-water; (d) 3.02-mm circular
capillary, air-ethanol; (e) 2.89-mm square capillary, air-ethanol; and (f) 2.89-mm square capillary, air-oil mixture.

coefficient (kLa) is expressed as correlations of Laborie et al.19 (eq 12) and Kreutzer2 and
Heiszwolf et al.40 (eq 11). Remarkably, an enormous
UTP1.19 amount of scatter is observed for the literature correla-
kLa ) 0.111 (13) tions. The Laborie et al. correlation results in a large
Lslug0.57 scatter distributed above and below the parity line,
whereas the Kreutzer-Heiszwolf et al. correlation
Inspection of eq 13 reveals that the relationship between largely underestimates the experimental data. A pos-
kLa, UTP, and Lslug can approximately be expressed as sible reason for the latter could be that the gas-liquid
UTP flow patterns in the monolith reactor that was used
kLa ∼ (14) differed greatly from those in the capillary setup used
xLslug in this study. Besides, the complexity of the liquid slug
length and its dependence on such parameters as the
kLa can, in turn, be correlated to the gas-phase and configuration of the nozzle likely have a large role in
liquid-phase Reynolds numbers. By replacing kLa with this discrepancy. Further research is thus needed to
the right-hand term in eq 14, and based on the regres- understand the effect of the gas-liquid feed system on
sion of experimental data, the following correlation was liquid slug length.
obtained: 3.4. Two-Phase Pressure Drop. Many methods
have been proposed for estimating the two-phase fric-
UTP tional pressure drop (∆Pf) in capillaries. One of these is
) 0.088ReG0.72ReL0.19 (15)
xLslug the Lockhart-Martinelli multiplier method. First pro-
posed by Lockhart and Martinelli,41 it involves the
Figure 9 shows plots of experimentally determined definition of a two-phase multiplier and a Lockhart-
liquid slug lengths at varying gas and liquid superficial Martinelli parameter. To estimate the frictional pres-
velocities, as well as predicted slug lengths using eq 15. sure drop using this method, a flow-regime-dependent
The predictions seem to be good, although noticeable constant that is often called the Chisholm parameter
deviations can be observed in a few cases. Notwith- also must be evaluated. From studies in a vertical
standing, the predicted slug lengths generally follow the capillary, Mishima and Hibiki30 reported that this
same trends as the experimentally measured values. In constant shows a capillary diameter dependence that
Figure 10, experimental dimensionless liquid slug lengths must be taken into consideration. Furthermore, many
in the Taylor flow regime are compared with predicted investigators have reported on the inability of the
values, based on the present study and literature Lockhart-Martinelli correlation to predict experimental
4892 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

Figure 9. Experimental and predicted liquid slug length (Lslug), as a function of superficial gas and liquid velocities: (a) 2-mm circular
capillary, air-water; (b) 3.02-mm circular capillary, air-water; (c) 2.89-mm square capillary, air-water; (d) 0.91-mm circular capillary,
air-ethanol; (e) 2-mm circular capillary, air-ethanol; and (f) 0.99-mm square capillary, air-ethanol. The closed shapes are experimental
data points, whereas the lines represent predicted values using eq 15.

is the fact that they are flow-regime-independent cor-

relations. This could, to a large extent, account for why
they have generally been observed to fail at low liquid
flow rates, where, provided an appreciable gas flow
ensues (such as in Taylor flow), significant deviation
from homogeneous flow occurs. Interestingly, however,
it is often at these low liquid flow rates that monolith
reactors would typically be operated, i.e., 0.05 < UG <
0.4 m/s, depending on the process in question. In view
of this, a flow-regime-dependent pressure drop model
would be developed based on data obtained from the
experimental study undertaken, beginning with a dis-
cussion of ∆Pf.
3.4.1. General Observations of Frictional Pres-
sure Drop. For co-current upward two-phase flow in a
vertical capillary, the frictional pressure drop can be
calculated from the measured total pressure drop, given
Figure 10. Comparison of experimental dimensionless liquid slug by
length (Ψslug,expt) with predicted values by various correlations
(Ψslug,pred). ∆PT ) ∆Pf + LFLgLc (16)
pressure drop data, especially at low liquid flow rates.29,42
Triplett et al.27 used the homogeneous pressure drop Figure 11a depicts the frictional pressure drop over
model to calculate ∆Pf, stating that predictions were the 1.4 m length of the 3.02-mm circular capillary for
good in the bubbly and Taylor flow regimes at high ReL, air-water system at different gas and liquid superficial
where, as claimed, the homogeneous flow assumption velocities. Two notable features can be observed from
is applicable. However, significant deviations were the figure: (1) pressure drop values are negative at
reported for slug-annular flow, annular flow, and Taylor lower liquid flow rates, which correspond to the Taylor
flow at relatively low ReL. One drawback of the Lock- flow regime; and (2) at higher liquid flow rates, the two-
hart-Martinelli and homogeneous pressure drop models phase frictional pressure drop increases as UG and UL
 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 4893

Figure 11. (a) Effect of superficial gas and liquid velocities on frictional pressure drop for the air-water system in the 3.02-mm circular
capillary. (b) Frictional pressure drop at zero net liquid flow condition for the air-oil mixture system in the 3.02-mm circular capillary.

increase, but, in the lower-liquid-flow-rate region, changes By comparing eqs 18 and 19, an equivalent velocity,
in the pressure drop do not seem to be regular. Similar with respect to the total pressure drop, may be defined
trends were observed for other systems that have been as

( )
investigated. Negative frictional pressure drops in two-
phase flows have scarcely been reported in open litera- dc2
ture. As can be seen in Appendix I of the Supporting Ue ) F g (20)
32µL L
Information accompanying this paper, a negative fric-
tional pressure drop was observed for the bulk of
In subsequent discussions, this velocity will be called
experimental data at very low liquid velocities. In an
the gravity-equivalent velocity. It can be considered as
attempt to understand this phenomenon, experiments
the liquid velocity in the capillary that would result in
were conducted under a zero net liquid flow condition,
a pressure loss equivalent to the hydrostatic pressure
i.e., liquid was fed batchwise into the capillary while
exerted by the liquid phase. Assuming laminar flow for
the gas feed was continuous. The result is shown in
both the gas and liquid phases, the two-phase gravity-
Figure 11b for an air-oil mixture in the 3.02-mm
equivalent velocity becomes
circular capillary. Observe that, over the range inves-

( )
tigated, all frictional pressure drops are negative. As
noted by Nicklin,43 a negative frictional pressure drop
dc2
Ue )  F g (21)
means that the total pressure drop is less than the 32µL L L
hydrostatic pressure drop, because slip between phases
can result in local down flows of liquid, resulting in wall and a two-phase mixture velocity UE, which is defined
shear stresses that act opposite to the usual sense. as the sum of the two-phase superficial velocity (UTP)
3.4.2. Theoretical Considerations. The apparently and the gravity equivalent velocity, is
complicated behavior of frictional pressure drop, as
evidenced previously, requires that, in the development UE ) UTP + Ue (22)
of a correlation for predicting the total pressure drop
∆PT, negative frictional pressure drop data is taken into A dimensionless two-phase pressure factor FE can
account. The extent to which two-phase pressure drops further be defined, analogous to the Fanning friction
are influenced by flow regimes, fluid properties, and factor of eq 5:
channel geometries also must be considered. With these
in mind, the following analysis is provided. Consider, ∆PT/Lc
FE ) (23)
for example, a single-phase vertical tube with liquid 1
/2FLUE2(4/dc)
flowing in the laminar regime. The total pressure drop
is the sum of frictional and static components and can 3.4.3. Correlation of Pressure Drop Data. In a
be represented as situation where both the gas and liquid-phase flows are
laminar, the pressure factor can be expected to take on
∆PT ) ∆Pf + FLgLc (17) a form similar to the Fanning friction factor, i.e.,
In this situation, the frictional pressure drop is given C
by the Hagen-Poiseuille equation: FE ) (24)
ReE
32µLULLc
∆Pf ) (18) where ReE, which is the modified Reynolds number, is
dc2 defined as

Substitution of eq 18 into eq 17 and rearranging yields FLUEdc

ReE ) (25)

[ ( ) ]
µL
2
32µLLc dc
∆PT ) UL + F g (19) Under this condition, the gas and liquid phases can be
dc2 32µL L
viewed as a homogeneous mixture. Based on the discus-
4894 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

homogeneous regime, the following analysis was per-

formed. A plot of S versus FE/(C/ReE) for UG/UL > 0.5 is
shown in Figure 13a, from which the following depend-
ence was deduced:

FE
∝ S-0.5 (26)
C/ReE

Through further data analysis, utilizing the parameter

FES0.5/(C/ReE) as shown in Figure 13b, the following
correlation was obtained:

Cf -0.5
Figure 12. UG/UL versus FE/(C/ReE), showing that, when UG/UL
FE ) S [exp(-0.02ReE) + 0.07ReE0.34] (27)
ReE
> 0.5, the two-phase flow deviates from the homogeneous regime.

sion of the slip ratio S presented previously, it was Equation 27 provides a convenient way of estimating
shown that homogeneous flow generally occurs when the nonhomogeneous regime pressure factor, which,
UG/UTP < 0.5, which is characterized by a slip ratio S when combined with eq 23, allows for the prediction of
close to unity. A slip ratio of S > 1 was shown to indicate the total pressure drop. Figure 13c illustrates the
deviation from homogeneous flow. Therefore, it can be comparison of the experimental and correlation-pre-
postulated that a relation between FE/(C/ReE) and UG/ dicted pressure factors, utilizing all experimental data.
UTP (or, equivalently, UG/UL) could allow for determi- Note that the predicted pressure factors for UG/UL <
nation of the transition from homogeneous to non- 0.5 were computed with eq 24, whereas eq 27 was used
homogeneous flow for pressure drop computations. The to compute the predicted pressure factors for UG/UL >
plot in Figure 12 shows such a relationship, from which 0.5. Figure 14 shows typical experimental and predicted
the following observations can be made: total pressure drop values. The predictions seem to be
(1) For UG/UL < 0.5, FE/(C/ReE) data approach a single very good, especially at low and moderate liquid veloci-
line that approximately corresponds to FE/(16/ReE) ) 1 ties, where Taylor flow occurs. It is also worthy to note
and FE/(14.2/ReE) ) 1 for the circular and square that it is at such low liquid velocities that the Lockhart-
capillaries, respectively, with an uncertainty of ap- Martinelli and homogeneous pressure drop models fail
proximately (9%, indicating that the gas and liquid to predict the pressure drop, verifying the need for a
phases can be approximately viewed as a homogeneous flow-regime-dependence approach to pressure drop es-
mixture. The converse is the case for UG/UL > 0.5, where timation.
significant deviation from homogeneous flow can be
observed. 4. Conclusions
(2) The capillary diameter and liquid viscosity were
observed to influence the parameter FE/(C/ReE) signifi- Two-phase flow hydrodynamics in vertical capillaries
cantly for UG/UL > 0.5. This is evidenced by comparing with circular and square cross sections that have
the data of the 0.91-mm, 2-mm, and 3.02-mm circular hydraulic diameters from 0.9 mm to 3 mm were experi-
capillaries for air-ethanol, as well as the data of the mentally studied, using air as the gas phase and water,
3.02-mm circular and 2.89-mm square capillaries for ethanol, or an oil mixture as the liquid phase. Flow
both the air-oil mixture and air-water system. regimes, bubble rise velocity, liquid slug length, and
Therefore, for homogeneous flow (UG/UL < 0.5), the pressure drop were investigated, with the gas and liquid
total pressure drop can be predicted using eq 24 to first superficial velocities being varied in the range of
estimate the pressure factor and eq 23 for computing 0.008-1 m/s. Based on the work performed and the
the pressure drop. To predict the total pressure drop discussions presented, the following major conclusions
when UG/UL > 0.5, wherein flows deviate from the can be drawn:

Figure 13. (a) Relationship between the slip ratio S and FE/(C/ReE) for UG/UL>0.5. (b) Correlation of experimental data for UG/UL > 0.5
for predicting the pressure factor (FE). (c) Comparison of pressure factors calculated from experimental pressure drop (FE,expt) data with
predicted pressure factors (FE,pred).
 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 4895

Figure 14. Variation of the experimental and predicted total pressure drop (∆PT) with varying superficial gas and liquid velocities: (a)
2-mm circular capillary, air-water; (b) 3.02-mm circular capillary, air-water; (c) 2-mm circular capillary, air-ethanol; and (d) 2.89-mm
square capillary, air-ethanol. The closed shapes are experimental data points, whereas the lines represent predicted values.

(1) Within the gas and liquid velocity range under posed based on the definition of the dimensionless
which experiments were conducted, four distinct flow pressure factor. Two pressure factor correlations were
regimes were observed: bubbly, slug-bubbly, Taylor, presented: one for homogeneous flow and the other for
and churn. A fifth regime, the annular flow regime, nonhomogeneous flow. Based on the analysis of experi-
occurred at excessively high gas and low liquid veloci- mental data, a value of UG/UL ) 0.5 was determined to
ties. indicate the transition point from homogeneous to
(2) The slip ratio S was determined to be a useful nonhomogeneous flow for pressure drop computation.
parameter for gauging the transition from homogeneous Very good pressure drop predictions were obtained as
to nonhomogeneous flow. It was demonstrated that, shown.
when UG/UTP < 0.5, S approaches a value of unity,
indicating homogeneous flow. For UG/UTP g 0.5, a Acknowledgment
significant increase in S was observed, indicating
significant deviation from homogeneous flow. Moreover, The Netherlands Foundation for Scientific Researchs
high S values occurred predominantly in the Taylor flow Chemical Sciences Division (NWO-CW) provides a
regime. research program subsidy. One of the authors (H.L.)
(3) The influences of capillary geometry, capillary thanks Beijing University of Chemical Technology,
hydraulic diameter, and fluid properties on bubble rise China, for financial support during this work.
velocity were determined to be of little significance.
(4) A new and simplified correlation for predicting Supporting Information Available: Hydrodynam-
bubble rise velocity and, by implication, the gas holdup ics of Taylor flow in vertical capillaries; flow regimes;
in vertical-capillary two-phase flow was proposed. bubble rise velocity; liquid slug length; and pressure
(5) A correlation for estimating the liquid slug length drop. Appendix I contains the raw experimental data
was developed and was satisfactorily able to predict the of this study, whereas Appendix II contains further
experimental liquid slug lengths obtained in this study. information on correlations for predicting bubble rise
However, the fact that existing literature correlations velocity (PDF). This material is available free of charge
showed an enormous amount of scatter and deviation, via the Internet at http://pubs.acs.org.
when compared to the correlation proposed in this
study, leaves open the question as to what extent the Notation
configuration of the gas-liquid nozzle (an experimental
setup dependent parameter) affects liquid slug lengths C ) constant relating the Fanning friction factor to laminar
in a given setup. flow Reynolds number (also relates the pressure factor
(6) For the prediction of the total pressure drop in a to the modified Reynolds number)
vertical-capillary two-phase flow, a method was pro- dc ) capillary hydraulic diameter (m)
4896 Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

fb ) bubble frequency (s-1) (10) Boger, T.; Zieverink, M. M. P.; Kreutzer, M. T.; Kapteijn,
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