Flow Pattern and Pressure Drop of Upward Two-Phase Flow in

Ind. Eng. Chem. Res. 2008, 47, 243-255

243

Flow Pattern and Pressure Drop of Upward Two-Phase Flow in Vertical Capillaries Dingsheng Liu†,‡ and Shudong Wang*,† Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People’s Republic of China, and Graduate School of Chinese Academy of Sciences, Beijing 100039, People’s Republic of China

Gas-liquid two-phase flow patterns were investigated visually with a digital camera for upward air-water flow in capillaries with diameters of 1.47, 2.37, and 3.04 mm. Several distinctive flow patterns were observed and described. The flow pattern maps developed in this work were compared with the conventional flowpattern transition criteria for larger-diameter tubes and the experimental data that have been published. The total pressure drop in capillaries was measured with a differential pressure transducer for Taylor flow, which existed broadly in multiphase monolith reactors and microchannel reactors. A new correlation was presented to calculate the total pressure gradient (TPG) of Taylor flow. In addition, the other three models, which were proposed previously by other researchers, were also used to predict the TPG. The four theoretical results all were in overall agreement with the experimental data. Finally, the effects of method used to calculate the void fraction on the TPG were checked. It was observed that the void fraction influenced the TPG remarkably, and the drift flux model was not suitable to evaluate the void fraction under the present experimental conditions. 1. Introduction Monolith honeycomb reactors are now widely considered to be a promising alternative to conventional gas-liquid-solid three-phase reactors, such as trickle-bed and slurry reactors.1-3 Generally, the monolith structure consists of an array of parallel, straight, uniform channels with square or circular geometry, typically having hydraulic diameters in the range of 1-5 mm.1-3 Fundamental knowledge of the two-phase flow in small channels, such as the flow pattern and the pressure drop, is, thus, demanded urgently for optimizing engineering design as well as evaluating practical performance. A pioneer systematic study on the characteristics of the gas-liquid flow in monolith reactors was performed by Satterfield and Ozel.4 They showed that the two-phase flow in a single channel of monolith reactors could be characterized by the two-phase flow in a capillary. 1.1. Flow Pattern. It is well-known that the morphology of gas-liquid flow greatly influences the rates of mass and heat transfer in small tubes. To this day, many papers related to flow patterns of multiphase flow in capillaries or microchannels have been published. Barnea et al.5 compared the experimental flow patterns from the capillaries with diameters on the order of millimeters, with the theory6 applied to the tubes with diameters on the centimeter scale. Barnea et al. considered surface tension to account for the deviation between the experimental data and the theoretical results. For capillaries, surface tension was a dominant factor, whereas, for larger diameter tubes, KelvinHelmholtz instability was a key factor.5 Mishima and Hibiki7 experimentally observed some flow regimes that were peculiar to capillaries, such as intermittent strings of bubbly flow. In addition, the boundaries between flow regimes were wellpredicted by the model of Mishima and Ishii.8 Coleman and Garimella9 considered hydraulic diameter and surface tension to be important factors in determining the locations of flow* To whom correspondence should be addressed. Tel.: +86-41184662365. Fax: +86-411-84662365. E-mail address: [email protected]. † Dalian Institute of Chemical Physics, Chinese Academy of Sciences. ‡ Graduate School of Chinese Academy of Sciences.

pattern transitions. Decreasing the capillary diameters shifted the transition to a dispersed flow pattern to a higher superficial liquid velocity, because of the combination of surface tensions and tube diameters. The transition to purely annular flow almost occurred at a constant superficial gas velocity and approached a limiting value as the capillary diameters decreased. Triplett et al.10 conducted an experimental investigation of two-phase flow patterns in horizontal circular capillaries. In their experiments, five major flow patterns were observed: bubbly, Taylor, churn, slug-annular, and annular flow. The data agreed well with some existing experimental data but poor consistency was observed with the predictions based on the relevant flow-pattern transition models. For vertical triangular capillaries with hydraulic diameters of 2.866 and 1.443 mm, Zhao et al.11 observed flow patterns similar to that encountered in the conventional large-sized vertical circular tubes. It was also determined that the transition boundaries from slug flow to churn flow and from churn flow to annular flow in the flow pattern maps shifted to the right as the capillary hydraulic diameters decreased. Chen et al.12 noticed, experimentally, a peculiar flow pattern that was called bubble-train slug flow. The flow pattern was characterized by the presence of a bubble train that was formed by several bubbles connected together with a membrane between two neighboring bubbles. The number of bubbles in a bubble-train slug appeared in an irregular manner, which was indicative of its inherent turbulent nature. Liu et al.13 studied two-phase flow using air and different liquids in circular and square capillaries. Four flow patterns, including bubbly, slug-bubbly, Taylor, and churn flow, were observed, under their experimental conditions. In addition another typical flow pattern, annular flow, occurred only at excessive high gas and liquid velocities. Chen et al.14 studied, experimentally, the effects of capillary diameters on vertical two-phase flow patterns. The flow-pattern maps showed that the transition boundaries of slug-churn flow and churnannular flow were dependent strongly on the capillary diameters. In contrast, the boundaries of dispersed bubble flow to churn flow and from bubbly flow to slug flow were less affected.

10.1021/ie070901h CCC: $40.75 © 2008 American Chemical Society Published on Web 12/01/2007

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Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008

Moreover, the experimental transition boundaries showed poor agreement with that predicted by existing models applied to larger tubes. 1.2. Pressure Drop. For two-phase flow in capillaries or microchannels, pressure drop is a very important parameter and is often used to design reactors and optimize operational conditions. Mishima et al.7 studied the frictional pressure drop of a gas-liquid two-phase flow in vertical capillaries using the Lockhart-Martinelli correlation15 and Chisholm’s equation.16 It was found that the value of the original Chisholm’s parameter (C) was dependent on not only the flow regimes but also the capillary diameters. The two-phase frictional pressure drop could be corrected well by Chisholm’s equation, in association with Chisholm’s parameter C, which is a function of the capillary diameter. Triplett et al.17 calculated the two-phase pressure drop in transparent, long, horizontal microchannels with circular and semitriangular cross-sections, using various two-phase friction models. For bubbly and slug flow patterns, the two-phase friction factor based on homogeneous mixture assumption provided the best agreement with the experimental data. For annular flow, the homogeneous mixture model and other correlations significantly overpredicted the frictional pressure drop. Zhao et al.18 reported the experimental data for the pressure drop of upward air-water flow in vertical miniature triangular channels. The results showed that two-phase frictional pressure drop could be well-predicted by the Lockhart-Martinelli method if their friction factor correlation for single-phase flow was adopted. Kawahara et al.19 measured and analyzed two-phase frictional pressure drop data in circular microchannels. In their experiments, the two-phase flow patterns were much less homogeneous, as indicated by video images and very large slip ratios. Thus, the agreements between the experimental data and the homogeneous flow model were generally poor, except for the good predictions (within (20%) obtained Dukler et al.’s function,20 which was used to calculate the mixture viscosity. In contrast, two-phase friction multiplier values were correlated well (within (10%) with the separated flow model of Lockhart and Martinelli15 when the coefficient C, which is used in Chisholm’s equation, was given by the model by Lee and Lee.21 Later, a conclusion similar to that of Kawahara et al.19 was also drawn by Yue et al.22 by investigating the pressure drop of twophase flow through T-type microchannel mixers. Recently, Liu et al.13 specifically studied the pressure drop of Taylor flow, one interesting and practical two-phase flow pattern in vertical capillaries. They correlated the dimensionless two-phase pressure factors and the total pressure drop using graVity-equiValent Velocities. To calculate the pressure drop of the homogeneous flow and nonhomogeneous flow, two different functions were developed to evaluate the pressure factors. In addition, the experimental data were predicted well by their theory, especially at low and moderate liquid velocities. Kreutzer et al.23 experimentally measured the total pressure drop of Taylor flow in a 2.3-mm vertical capillary, using a technique that independently controlled the lengths of liquid slugs and gas bubbles. They proposed a new method to calculate the total pressure drop by means of the lengths of liquid slugs. More recently, Akbar et al.24 simulated Taylor flow in capillaries based on the Volume of Fluid technique. Similar to Kreutzer et al.,23 they obtained a new function, which was used to calculate the frictional pressure gradient. The agreements between the simulation and the experimental data were very good. This review shows that many works about the two-phase flow patterns in capillaries have been conducted. However, rather inconsistent results and conclusions have also been obtained

by various researchers. This confusion may partially stem from varieties of nomenclatures defining flow-pattern transition boundaries or even flow patterns themselves. Generally, there is one undoubted fact: the behavior of fluid in capillaries is different from that in conventional larger tubes, because of size effects. It is significant to understand the physical mechanisms of flow patterns further, to enhance the performances of multiphase monolith reactors and microchannel reactors. The aforementioned review also indicates that many methods have been proposed to calculate the two-phase pressure drop in capillaries or microchannels. Most of them are independent of two-phase flow patterns, which make the methods inapplicable under certain circumstances. Therefore, it is not puzzling to observe significant deviations from experimental data without knowing specific flow regimes. Taylor flow is a very common and important flow pattern in multiphase monolith reactors. Although some researchers, including Liu et al.,13 Kreutzer et al.,23 and Akbar et al.,24 have reported some new methods and data about pressure drop of Taylor flow in capillaries, there is no universal theory yet and more data on the pressure drop in Taylor flow must be accumulated. Therefore, the work performed here was designed to systematically study the flow patterns and pressure drop of the gasliquid flow in vertical capillaries, which were similar to the channels of the monolith reactors. The experimental results were compared to the available relevant data and theories. 2. Experiment The experimental setup of the upward gas-liquid flow in vertical capillaries was clearly demonstrated in Figure 1. As shown in Figure 2, the inlet part of the capillary was connected to a T-shaped mixer with the same internal diameter to the capillary. The outlet part of the capillary was connected to a straight junction with the same structure as the T-mixer, but without the side liquid inlet. Deionized water and air were chosen for the liquid phase and the gas phase, respectively. To force liquid through the capillaries at a constant velocity, a tank with invariable pressure was designed to supply water. The water velocity was adjusted and measured by a liquid rotameter between the water bank and the T-mixer. The air velocity was adjusted by a mass flow controller. Water and air were fed into the capillaries from the side and bottom of the T-mixer, respectively. A separator was used to drain water and discharge air at the exit of the system, where the pressure was atmospheric. Three different circular quartz-glass capillaries were used to investigate flow patterns and pressure drop. The capillary internal diameters were determined by imaging the capillary cross-sections using a microscope. The physical parameters of the capillaries are given in Table 1. The flow patterns were captured by a digital charge-coupled device (CCD) camera with the shutter speed of 1/1000-1/4000 s and then were saved onto the hard drive of a personal computer (PC) for later analysis. The pressure drop along the capillaries was measured with a differential pressure transducer (DPT). As shown in Figure 1, the two sampling ports of the DPT were connected to the side ports of the T-mixer and the straight junction, respectively. Figure 2 shows that the distance from the T-mixer’s side port to the capillary inlets was only 2 mm, so it was negligible, in regard to the local effect of the T-mixer on the total pressure drop, with respect to the total lengths of the capillaries. The same conclusion was suited for the capillary outlets. 3. Results and Discussion 3.1. Two-Phase Flow Pattern. 3.1.1. Flow Pattern. As shown in Figures 3 and 4, different flow patterns were observed

Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 245

Figure 1. Schematic diagram of the experimental setup.

Figure 2. Detailed structure of the T-mixer. Table 1. Specifications of the Capillaries Used in the Experiment inner diameter (mm)

length (m)

capillary number

1.47 2.37 3.04

0.761 0.917 0.834

1 2 3

at different superficial gas and liquid velocities for upward gasliquid flow in the vertical capillaries. To determine flow patterns clearly and eliminate confusion of terminologies used by various researchers, the specific definitions for different flow patterns are given as follows: Bubbly flow: Bubbly flow often occurs when superficial gas velocities are low while superficial liquid velocities are high.

Figure 3. Representative flow patterns in the 1.47-mm capillary at a low superficial liquid velocity: (a) UG ) 0.0113 m/s, UL ) 0.0786 m/s, bubbly flow; (b) UG ) 0.3048 m/s, UL ) 0.0786 m/s, Taylor flow; (c) UG ) 1.2203 m/s, UL ) 0.0786 m/s, bubble-train slug flow; (d) UG ) 4.4745 m/s, UL ) 0.0786 m/s, churn flow; and (e) UG ) 11.0351 m/s, UL ) 0.0786 m/s, annular flow.

It is characterized by distinct or distorted sphere bubbles; generally, the diameters of the bubbles are less than or equal to the capillary inner diameters. Taylor flow: Taylor flow (which is also called slug flow, bubble train flow or segmented flow) is characterized by long bubbles separated by liquid slugs. The lengths of the bubbles are greater than the capillary inner diameters. Thin liquid films exist between the bubbles and the capillary walls. Slug-bubbly flow: This flow pattern simultaneously has the characteristics of Taylor flow and bubbly flow. The flow pattern

246


Figure 4. Representative flow patterns in the 1.47-mm capillary at a high superficial liquid velocity: (a) UG ) 0.0113 m/s, UL ) 1.6365 m/s, bubbly flow; (b) UG ) 0.0727 m/s, UL ) 1.6365 m/s, bubbly flow; (c) UG ) 1.2203 m/s, UL ) 1.6365 m/s, bubbly flow; (d) UG ) 2.3259 m/s, UL ) 1.6365 m/s, slug-bubbly flow; (e) UG ) 4.4745 m/s, UL ) 1.6365 m/s, bubbletrain slug flow; and (f) UG ) 11.0351 m/s, UL )1.6365 m/s, churn flow.

is more disordered and is often considered to be a transition flow pattern. Bubble-train slug flow: This flow pattern typically consists of series of bubbles, similar to trains, with a clear interface between the connecting bubbles.12 The number of bubbles in a “train” seems to appear in a random manner. In addition, the shape and size of bubbles are not always uniform, under certain circumstances. Churn flow: This flow occurs when superficial gas velocities are large enough for the successive several bubbles to coalesce to one bubble after breaking through the liquid slugs between them. A wave or ripple motion is often observed at the bubble tail with tiny gas bubbles entrained in the liquid slug because of the high superficial gas velocity. To some extent, churn flow is similar to Taylor flow, but the former is more chaotic and disordered. Annular flow: Annular flow is observed at excessively high superficial gas velocities and very low superficial liquid velocities. This flow is comprised of a continuous gas phase in the central core and a continuous liquid phase that is deposited on the circumference of the capillary walls. One noteworthy feature is that the major flow patterns, classified as given previously, can be further subdivided into more-specific flow patterns. In other words, some slight difference in appearance can be observed for the flow pattern with the same name. For example, Figure 4a-c showed the diversity of appearance of bubbly flow due to different superficial gas velocities. Figures 3 and 4 show typical images of the two-phase flow observed in the 1.47-mm capillary at low and high superficial liquid velocities, respectively. With increasing superficial gas velocities, five distinct flow patternssincluding bubbly flow, Taylor flow, bubble-train slug flow, churn flow, and annular flowsappeared one after another at a constant low superficial liquid velocity in Figure 3. Moreover, these flow patterns were comparative ordered or regular. In contrast, at a high superficial liquid velocity in Figure 4, every flow pattern appeared much more disordered and chaotic. In addition, bubbly flow, slugbubbly flow, bubble-train flow, and churn flow were recorded in a sequence with increasing superficial gas velocities. Furthermore, the comparison of the flow patterns between the low and high liquid velocities were performed. Bubbles of the bubbly

flow in Figure 3 were spherical or spheroidal, and the distances between the bubbles were uniform. Bubbles confined in the capillary rose straight regularly. Figures 4a-c show that the number of bubbles in the bubbly flow increased as the gas velocities were enhanced. Also, the bubbles moving up along the capillary oscillated from one side of the wall to another side, which was similar to the phenomenon noticed by Krishna et al.25 In addition, slug-bubbly flow was not observed at the low superficial liquid velocity, and Taylor flow and annular flow were not noticed at the high superficial liquid velocity. 3.1.2. Flow Pattern Map. Figures 5a, 5b, and 5c show the overall flow pattern maps for the three circular capillaries (with diameters of 1.47, 2.37, and 3.04 mm), where the superficial liquid and gas velocities were used as the ordinate and abscissa, respectively. The solid lines represented the boundaries at which flow pattern transitions occurred. Four major flow patternss including bubbly flow, Taylor flow, churn flow, and annular flowsdivided the flow pattern maps into four regions, where the flow patterns possessed definite characteristics that were easily observed. The two transition flow patterns, slug-bubbly flow and bubble-train slug flow, were situated in the small central transition regions of the flow pattern maps. To predict the pressure drop of Taylor flow exactly, which will be described in section 3.2, quasi-Taylor flow was presented as an independent flow pattern only in Figure 5. Here, quasi-Taylor flow was defined as the Taylor flow of which the lengths of liquid slugs were smaller than or equal to capillary diameters. The flow patterns in the transition regions represented ambiguity characteristics and were judged more subjectively. Because these measurements were on the same order of size for the capillary diameters in this study (millimeters), the effects of diameter on flow patterns were not remarkable. Moreover, the three flow pattern maps in Figure 5 were similar (overall). It is well-known that flow patterns are dependent on not only liquid velocities but also gas velocities. In other words, if the gas (or liquid) velocity is not appropriate, some specific flow patterns will not appear at any liquid (or gas) velocity in Figure 5. Therefore, the flow pattern maps are helpful in determining the operational conditions in industry. 3.1.2.1. Comparison with Existing Transition Correlations. The present experimental data were compared with two famous models that have been developed to predict flow pattern transitions during steady gas-liquid flow in vertical circular tubes: one by Taitel et al.6 and another by Mishima et al.8 To predict the operational conditions under which the transitions will occur, it is very important to understand the physical mechanisms of flow transitions. However, considerable disagreements among various transition mechanisms still exist. Therefore, the theories with different physical mechanisms sometimes predicted different flow patterns under the same operational conditions. In 1980, Taitel et al.6 presented a physical model and developed theoretically based transition equations as follows. For the transition from bubbly flow to Taylor flow, Taitel et al.6 gave an expression for the flow at which dispersed bubbles were observed:

UL ) -UG + 4.0

{

[

] }

d0.429(σ/FL)0.089 g(FL - FG) FL ν 0.072 L

0.446

(1)

This function was plotted in Figure 6 as curve A; above this curve, bubbly flow existed. However, regardless of the amount of liquid turbulent energy for breaking and dispersing the gas phase, bubbly flow could not exist at a gas void fraction


Figure 5. Flow pattern maps for upward gas-liquid flow in vertical capillaries: (a) d ) 1.47 mm, (b) d ) 2.37 mm, and (c) d ) 3.04 mm.

of R ) 0.52, which was represented as beeline B in Figure 6. Here,

R)

UG UG + UL

(2)

Thus, curve A, which delimited bubbly flow, terminated at beeline B. Bubbly flow existed in the regions to the left of beeline B and on curve A. Below curve A, Taylor flow dominated. Increasing the gas velocities further would shift

Figure 6. Comparison of the experimental flow pattern maps with Taitel et al.’s model:6 (a) d ) 1.47 mm, (b) d ) 2.37 mm, and (c) d ) 3.04 mm.

Taylor flow to churn flow. The criteria of the transition between Taylor flow and churn flow was expressed as

(

LE UG + UL ) 40.6 + 0.22 d xgd

)

(3)

where LE was the entrance length of the tube. In this work, LE represented the length from the entrance to the midpoint of the capillary that was used. Equation 3 was denoted by curve D.

248


The transition criteria were plotted as vertical beeline C in Figure 6. The regions to the right of beeline C were dominated by annular flow. (Note that the flow pattern names shown in the brackets in Figures 6- 8 are those given by the respective authors.) As seen from Figure 6, the theoretical predictions are not consistent with the experimental results completely. As mentioned previously, bubble-train slug and slug-bubbly are two transition flow patterns. In Figure 6, beeline B and curve D were within or near the transition regions, which meant the theory predicted the transition from bubbly flow to churn flow and the transition from Taylor flow to churn and annular flow well. For the capillaries with diameters of 2.37 and 3.04 mm, the correct locus of beeline E also meant good predictions. Discrepancies between the theory results and the experimental data existed, but the theoretical boundaries between bubbly flow and Taylor flow were still helpful to identify bubbly flow above curve A correctly. From Figure 6, the theoretical beeline C poorly predicted the transition from churn flow to annular flow, which might result from the fact that the simple criteria (eq 4) did not take into account the effects of liquid velocities and capillary diameters on flow patterns. In 1984, Mishima et al.8 considered the void fraction (R), as a simple and reliable parameter to predict flow pattern. From this point of view, one new flow pattern transition criterion for upward gas-liquid flow in vertical tubes was developed. The transition boundary of bubbly flow to Taylor flow was given by R ) 0.3, which was converted to a conventional form based on superficial velocities:

UL )

(

)

[

]

0.76 σg(FL - FG) 3.33 - 1 UG C0 C0 FL2

1/4

(5)

where

C0 ) 1.2 - 0.2

x

FG FL

(6)

Equation 5 was plotted as curve A in Figure 7. The regions to the left of the curves marked A were dominated by bubbly flow, and the proximate zones to the right of the curves marked A were controlled by Taylor flow. The transition from Taylor flow to churn flow was postulated to occur when

R g Rm

(7)

Here, R was calculated based on the drift-flux model

UG

R)

C0(UG + UL) + 0.35

(8)

x

∆Fgd FL

Figure 7. Comparison of the experimental flow pattern maps with Mishima et al.’s model:8 (a) d ) 1.47 mm, (b) d ) 2.37 mm, and (c) d ) 3.04 mm.

and Rm represents the mean void fraction of Taylor bubbles and was expressed as follows:

Moreover, Taitel et al.6 considered the fact that churn flow could be expected only at R > 0.25. Thus, the locus of R ) 0.25 is shown as dashed beeline E in Figure 6 and represents the terminus of curve D. For higher gas velocities, the flow became annular. The transition boundary between churn flow and annular flow was given by

Rm ) 1 - 0.813 ×

UGFG1/2 [σg(FL - FG)]1/4

) 3.1

(4)

{

(C0 - 1)(UG + UL) + 0.35 x∆Fgd/FL

}

(UG + UL) + 0.75 x∆Fgd/FL[∆Fgd3/(FLνL2)]1/18

0.75

(9)

The transition for the Taylor flow to churn flow occurred at the curves marked B, based on eq 7. The curves marked B intersected the beelines that were marked C and formed the churn-flow regions, which were below curve B and to the left of beeline C. At higher liquid and gas velocities, the Taylor


flow was limited by beeline D. At gas velocities above beeline D, liquid slugs would be dispersed into liquid droplets by entrainment; thus, annular flow should be observed. For two different mechanisms for occurrence of annular flow, Mishima et al. derived two transition equations as follows:8

UG )

x

∆Fgd (R - 0.11) FG

(10)

where R should satisfy the condition given by eq 7:

UG g

( )[ σg∆F FG2

1/4

µL

]

(FLσ xσ/(∆Fg))1/2

-1/5

(11)

Equations 10 and 11 were designated as beelines C and D, respectively. Beelines C and D intersected curve B at two different points. The three curves also consisted of a zone that was dominated by annular flow. Generally, wide discrepancies between the Mishima et al.8 theory and this study are observed in Figure 7. The transition boundaries from bubbly flow to Taylor flow was dependent only on the properties of both phases; they were not affected by the capillary diameters. In other words, the loci of the three curves marked A in Figure 7 were completely the same. From the experimental data, the transition boundaries from bubbly flow to Taylor flow were inclined to be parallel to the abscissas, while the theoretical curves marked A were vertical to the abscissa in the mass. Another experimental fact was that bubbly flow existed infrequently in the regions to the right of the curves marked A, except for the 1.47-mm capillary. The Taylor-flow regions predicted by the theory actually included several different flow patterns observed in the experiment. The agreement between the theory and the experiment was good only at the lower liquid velocities. With further increases in the gas velocities, the theoretical transition boundaries showed that churn flow and annular flow occurred at low and high liquid velocities, respectively. The prediction zones for churn flow just fell in the experimental transition region, including mainly bubble-train slug flow, partial Taylor flow, annular flow, and churn flow. The theoretical annular flow region was apparently the combination of the experimental churn and annular flow zones. That meant that, for any liquid velocity, the flow pattern would always be annular flow when the gas velocity was larger than the critical value. Obviously, the theoretical results for annular flow were proved incorrect by the experimental data in Figure 7. 3.1.2.2. Comparison with Existing Data. The flow pattern map for the 1.47-mm capillary was also compared with the existing experimental data of Triplett et al.10 and Zhao et al.11 in Figure 8. Triplett et al.10 distinguished five flow patterns in a horizontal circular capillary that had a diameter of 1.45 mm: bubbly, Taylor, slug-annular, churn, and annular flow. Zhao et al.11 observed bubbly, Taylor, churn, and annular flow in the vertical triangular capillary with a diameter of 1.44 mm. Figure 8a shows that Triplett et al.’s experimental data agreed well with the overall results determined in this work, except for the Taylor-flow region. The slug-annular flow pattern defined by Triplett et al.10 was assigned to annular flow here. The Taylor flow observed by Triplett et al.10 had a wider distribution than that in this work. The bubbly- and Taylor-flow regions observed by Zhao et al.11 in Figure 8b were similar to that recorded by Triplett et al.10 in Figure 8a. That made the bubbly-flow region smaller and the Taylor-flow region bigger than that in this work. The churn-flow region of Zhao et al.11 extended to lower liquid

Figure 8. Flow-regime map comparison of the present experimental data in 1.47-mm capillary with published experimental data: (a) Triplett et al.10 and (b) Zhao et al. 11

velocities but disappeared in trends at higher gas velocities. This distribution of churn flow was different from the results of the present work and Triplett et al.10 In Figure 8b, for any liquid velocity, the annular flow always occurred at higher gas velocities, which was inconsistent with the experimental data. 3.2. Two-Phase Pressure Drop. It is well-known that the total pressure gradient (TPG) of two-phase flow consists of three components, as follows:

dP dP dP [dX ] ) [dX ] + [dXdP] + [dX ] T

F

A

S

(12)

where [dP/dX]T, [dP/dX]F, [dP/dX]A, and [dP/dX]S represent TPG, the frictional pressure gradient (FPG), the acceleration pressure gradient (APG), and the static pressure gradient (SPG), respectively. During the present experiment, the pressure drop along the capillaries is very small, in comparison to atmosphere pressure, and no phase changing occurs, so the acceleration pressure drop can be negligible. Then, eq 12 becomes

dP dP dP [dX ] ) [dX ] + [dX ] T

F

S

(13)

As discussed previously, the pressure drop of only Taylor flow was evaluated. The long bubbles and liquid slugs are distributed alternately and orderly in Taylor flow. The liquid

250


film between the bubble and capillary wall is so thin that the weight of the liquid film can be negligible. Because of the special characteristics of Taylor flow, the SPG and the void fraction can be calculated using eqs 14 and 2, respectively.

dP [dX ] ) (1 - R)F g L

S

(14)

3.2.1. Correlation of the TPG. The total pressure drop of Taylor flow results from the combination of gas bubbles and liquid slugs. The shearing forces on liquid film exerted by gas bubbles are so small that they can be considered to be negligible. Because of gravitation, the liquid film around bubbles flows downward along the capillary wall. The upward frictional force acting on the film is also negligible for very low velocities of the film. Thus, the total pressure drop can be assumed to result from only liquid slugs. In addition, it is postulated that the liquid flow is fully developed in each liquid slug. For the present Taylor flow, ReL < 1000, which means that the flow is laminar in each liquid slug. Consequently, the two-phase frictional pressure drop is given by the Hangen-Poiseuille formula:

∆PF )

32µLVLLc(1 - R) d2

(15)

Here,

VL )

UL 1-R

(16)

Thus, eq 15 is expressed as

dP [dX ]) F

32µLUL d2

(17)

In Taylor flow, the longer the liquid slugs are, the more fully the liquid flow develops. Based on the assumptions presented previously, eqs 15-17 are less-suited for quasi-Taylor flow, as defined in section 3.1.2. Therefore, the pressure drop of quasiTaylor flow was not considered here. From Figures 5a-c, it can be observed that the quasi-Taylor flow zone is very wide for the 1.47-mm capillary, while being relative small for the capillaries with diameters of 2.37 and 3.04 mm. Hence, the TPG in the two bigger-diameter capillaries were calculated using eqs 2, 13, 14, and 17 and were plotted against the experimental data in Figure 9. Note that, for every constant superficial liquid velocity, the superficial gas velocity decreases in the positive direction of the abscissas in Figure 9. The agreement between the theoretical results and the experimental data was good in the mass. As observed in the experiment and judged in Figure 5b, quasi-Taylor flow disappeared quickly toward the bottomleft. This fact was reflected in Figure 9a that the discrepancy between the theory and the experiment became smaller toward the bottom-left direction. The same phenomenon was also observed in Figure 9b, but was relative inconspicuous to that in Figure 9a, which might be because that the FPG resulted from liquid slugs was very small in the 3.04-mm capillary. In addition, the values of the FPG were estimated by several other methods, including a homogeneous-flow model, a separatedflow model, and the flow model of Liu et al.13 Different from eq 13, Liu et al.13 deduced a new method to calculate the TPG of Taylor flow in capillaries without knowing the values of FPG

Figure 9. Comparison of the experimental total pressure gradient (TPG) with the predictions based on eqs 2, 13, 14, and 17.

and SPG in advance. The present experimental data were compared with the methods mentioned in the following sections. 3.2.2. Comparison with the Homogeneous-Flow Model. In the homogeneous-flow model, it is postulated that two-phase flow can be treated as a hypothetical single-phase flow that has some type of average or uniform properties.29 In this section, the reliability of the homogeneous-flow model, in regard to predicting the TPG, was examined. Generally, in the homogeneous flow model, FPG is calculated from

1 G2 d 2FTP

( ) ( )( )

(18)

1 m 1-m ) + FTP FG FL

(19)

dP dX

)f

F

where

The two-phase friction factor (f) is a function of the homogeneous Reynolds number, ReTP:

ReTP )

Gd µTP

(20)

For laminar flow,

f)

64 ReTP

(21)

Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 251 Table 2. Expressions for the Two-Phase Mixture Viscosity two-phase viscosity model

function

number

McAdams26

1 m 1-m ) + µTP µG µL

Dukler et al.20

µTP )

Cicchitti et al.27

µTP ) mµG + (1 - m)µL

Lin et al.28

µTP )

1

( ) ( )

UG UL µ + µ U G U L

2 3

µLµG

4

µG + m1.4(µL - µG)

Table 3. Chisholm Parameter (C) Values for Various Flow Combinations liquid-flow regime

gas-flow regime

Chisholm parameter, C

viscous turbulent viscous turbulent

viscous viscous turbulent turbulent

5 10 12 20

and ReTP is dependent on the two-phase mixture viscosity (µTP). As shown in Table 2, several two-phase viscosity models are used to evaluate µTP. It is obvious that FPG (or, consequently, the TPG) is influenced by the choice of the two-phase viscosity models. The experimental TPG were compared with the predictions based on the homogeneous-flow model with eqs 2, 13, 14, and 18 in Figure 10. This figure shows that the agreement between the experimental data and the homogeneous-flow model was generally good for the four viscosity models. However, the error increased to beyond (30% at lower TPG values, except for the viscosity model of Dukler et al.20 At moderate TPG, the viscosity model of Dukler et al.20 underpredicted the values, especially for the 2.37-mm capillary. From Figure 10, the theoretical predictions approached the experimental data for the 3.04-mm capillary better than those for the 2.37-mm capillary. 3.2.3. Comparison with the Separated-Flow Model. In the separated-flow model, the phases are treated as if they are separated and flow in well-defined but unspecified parts of the cross section.15,29 The two-phase FPG is calculated using the two-phase multiplier Φ2L that was defined by Lockhart and Martinelli15 as follows:

dP [dX ] ) Φ [dXdP] 2 L

F

L

(22)

Here, [dP/dX]L represents the FPG when only the liquid is assumed to flow in the capillary. The two-phase multiplier is correlated with the Lockhart-Martinelli parameter (X2), which is the ratio of the single liquid and gas FPG:

X2 )

(dP/dX)L (dP/dX)G

(23)

where (dP/dX)G denotes the FPG when only the gas is postulated to flow in the capillary. The correlation between Φ2L and X2 can be represented well by the equation

Φ2L ) 1 +

C 1 + X X2

(24)

where C is the Chisholm parameter. The values of C, depending on whether each phase was in turbulent-flow or laminar-flow conditions, for the various flow combination are shown in Table 3. For the present Taylor flow, ReLS < 500 and ReGS < 500. Therefore, the value of C would be 5, because both phases had

Figure 10. Comparison of the experimental TPG with predictions based on the homogeneous-flow model.

laminar flow. The C values were independent of tube diameters in Table 3. However, Mishima and Hibiki7 found that the value of the Chisholm parameter decreased as the tube diameters decreased. Therefore, they presented a new correlation:

C ) 21[1 - exp(-0.319d)]

(25)

Here, the diameter d is given in units of millimeters. However, the effect of the flow pattern on the Chisholm parameter C was not considered in eq 25. Based on the separated-flow model, the theoretical results, obtained with eqs 2, 13, 14 and 22, were plotted against the experimental data in Figure 11. According to Table 3 and eq 25, the different values of C were adopted. From Figure 11, the prediction values of TPG increased as the value of the parameter C increased, especially for the 2.37-mm capillary. Two separated-flow models that were conducted using different C values all agreed well with the experimental data. 3.2.4. Comparison with the Model of Liu et al.13 Recently, a Taylor-flow-dependent pressure-drop model was developed by Liu et al.,13 based on data obtained from their experimental study. In the model, one concept, called gravity-equivalent velocity, was presented. Gravity-equivalent velocity was regarded as the liquid velocity that would result in a pressure drop equivalent to the hydrostatic pressure drop exerted by the liquid phase.13 Assuming that both the gas phase and the liquid

252


Figure 11. Comparison of the experimental TPG with the predictions based on the separated-flow model.

phase had laminar flow, the two-phase gravity-equivalent velocity was defined as follows:

Ue )

( )

d2 (1 - R)FLg 32µL

(26)

and a two-phase mixture velocity (UE) was defined as the sum of the two-phase superficial velocity (UTP) and the gravityequivalent velocity (Ue):

UE ) UTP + Ue

(27)

The two-phase TPG was represented as

dP [dX ] ) 21(F U )(d4)F 2

L

T

E

E

(28)

where FE is a dimensionless two-phase pressure factor. Liu et al.13 considered that homogeneous flow occurred when UG/UL < 0.5, and FE was expressed as

FE )

16 ReE

(29)

while nonhomogeneous flow resulted when UG/UL > 0.5, and FE then was calculated as follows:

FE )

( )

16 -0.5 S [ exp(-0.02ReE) + 0.07ReE0.34] (30) ReE

Figure 12. Comparison of the experimental TPG with predictions based on the Liu et al. model.13

where ReE, which represents the modified Reynolds number, is defined as

ReE )

FLUEd µL

(31)

Comparisons of the experimental TPG with the predictions of the Liu et al. model13 are shown in Figure 12. It was observed that, at higher TPG values, the agreement between the theoretical predictions and the experimental data was good for both types of capillaries. However, a bigger deviation was observed at lower TPG values for the 2.37-mm capillary and at moderate TPG values for the 3.04-mm capillary. 3.2.5. Effects of the Methods Used To Calculate the Void Fraction on the TPG Value. Figures 9-12 show that the theoretical predictions, based on the four different methods, all agreed generally well with the experimental data. As mentioned previously, the several theoretical TPG values were all calculated using eqs 2, 13, and 14, except for the Liu et al.13 model. The TPG was evaluated without knowing the FPG and SPG in advance in the Liu et al.13 model. However, the void fraction (R) was also calculated in Figure 12 using eq 2. The SPG, calculated directly from R, seriously affected the TPG; therefore, the R values were very important for predicting the TPG exactly. The effects of the method used to calculate R on the TPG are clearly shown in Figure 13. In this figure, the R values were calculated using eqs 2 and 8 and are marked as the red and


Figure 13. Effects of method calculating the void fraction on TPG (R). Red symbols represent R values calculated based on eq 2; black symbols represent R values calculated based on eq 8.

black symbols, respectively. Note that, in Figure 13, the SPG, which is denoted by hollow circle symbols, was regarded as TPG, neglecting FPG in eq 13. Different void fractions resulted in different SPG and, consequently, different TPG. Except for the Liu et al.13 model, the other three theoretical results all overlapped with the SPG. In other words, the SPG remarkably determined the TPG in the three models. Thus, the method used to evaluate void fraction is very important. As shown in Figure 13, the drift flux model brings great deviation and is not suitable for predicting R in the present experiment. The method used to evaluate R, based on eq 2, is very good, and the errors produced are small, overall. Another factor is that the Liu et al.13 model is less influenced by the SPG, although the gravity-equivalent velocity Ue in eq 26 also is dependent on the void fraction. 4. Conclusions The upward two-phase flows in the vertical capillaries with diameters of 1.47, 2.37, and 3.04 mm were investigated, using

air as the gas phase and water as the liquid phase. The flow pattern maps were developed and compared with the existing data and theories. Two-phase pressure drop was measured, and then the TPG were calculated and analyzed by comparison with the existing models. Based on the present experiment and analyses, the following conclusions are drawn here: (1) Six distinctive flow patternssincluding bubbly, slugbubbly, Taylor, bubble-train slug, churn, and annular flows were observed during the experiment. The three flow pattern maps, based on the 1.47-, 2.37-, and 3.04-mm capillaries, were generally similar. The effects of capillary diameter on the flow patterns were not remarkable in this work. (2) The flow pattern transition criteria of Taitel et al.6 and Mishima et al.,8 based on larger tubes, were not consistent with the present experimental data completely. Taitel et al.’s theory predicted the transition from Taylor flow to churn flow and the transition from bubbly flow to churn flow well.6 Wide discrepancies between the Mishima et al. model and the experimental data were observed. Mishima et al.’s theory predicted the Taylor

254


flow zone correctly (but not precisely) only at low liquid velocities.8 Therefore, new theories suited for capillaries must be further investigated and developed. (3) Triplett et al.’s10 research results agreed well with the present work. However, the agreement between the present experiment data and Zhao et al.’s11 was not good, possibly because of the different capillary cross-sectional geometries. The accumulation of more experimental data is still necessary to identify flow patterns correctly and construct new theories. (4) One new simple correlation based on eqs 2, 13, 14, and 17, which are used to calculate the total pressure gradient (TPG) of Taylor flow, was proposed under liquid laminar flow conditions. The calculation results agreed well with the experimental data. It was proved that the effects of liquid slugs on TPG were crucial, whereas that of gas bubbles could be negligible. Moreover, the longer the liquid slugs of Taylor flow were, the more exactly the correlation was predicted. (5) The homogeneous-flow model, the separated-flow model, and the model reported by Liu et al.13 were compared with these experimental data, and the agreements were generally good. The method used to calculate the void fraction (R) was very important for predicting the TPG exactly. The homogeneousflow model, the separated-flow model, and the correlation proposed in this paper were seriously influenced by R, because the same procedure was used to calculate the static pressure gradient (SPG). However, the dependence of the Liu et al. model on R was smaller. The drift flux model was not suitable for evaluating R, whereas eq 2 worked very well in this work. To some extent, the SPG determined the TPG for Taylor flow. Nomenclature C ) Chisholm parameter C0 ) distribution parameter d ) capillary diameter (m) f ) two-phase friction factor FE ) pressure factor g ) acceleration of gravity (m/s2) G ) total mass flux (kg/(m2 s)) Lc ) length of capillary (m) LE ) entrance length of tube (m) m ) mass fraction ∆PF ) two-phase frictional pressure drop (Pa) ReE ) modified Reynolds number; ReE ) FLUEd/µL ReGS ) gas superficial Reynolds number; ReGS ) FGUGd/µG ReL ) Reynolds number for liquid slug in Taylor flow; ReL ) FLVLd/µL ReLS ) liquid superficial Reynolds number; ReLS ) FLULd/µL ReTP ) homogeneous Reynolds number; ReTP ) Gd/µTP S ) slip ratio; S ) (UG/R)/[UL/(1 - R)] Ue ) gravity-equivalent velocity (m/s) UE ) two-phase mixture velocity; UE ) UTP + Ue (m/s) UG ) superficial gas velocity (m/s) UL ) superficial liquid velocity (m/s) UTP ) two-phase superficial velocity (m/s) VL ) liquid-phase velocity (m/s) X2 ) Martinelli parameter Greek Symbols R ) void fraction Rm ) mean void fraction of the Taylor bubble Φ2L ) two-phase multiplier µG ) gas kinetics viscosity (kg/(m s)) µL ) liquid kinetics viscosity (kg/(m s)) µTP ) two-phase mixture kinetics viscosity (kg/(m s))

νL ) liquid kinematic viscosity (m2/s) FG ) gas density (kg/m3) FL ) liquid density (kg/m3) FTP ) two-phase mixture density (kg/m3) ∆F ) density difference; ∆F ) FL - FG (kg/m3) σ ) surface tension (N/m) Literature Cited (1) Kreutzer, M. T.; Kapteijn, F.; Moulijn, J. A.; Heiszwolf, J. J. Multiphase monolith reactor: Chemical reaction engineering of segmented flow in microchannels. Chem. Eng. Sci. 2005, 60, 5895. (2) Boger, T.; Heibel, A. K.; Sorensen, C. M. Monolithic catalysts for the chemical industry. Ind. Eng. Chem. Res. 2004, 43, 4602. (3) Roy, S.; Bauer, T.; Al-Dahhan, M.; Lehner, P.; Turek, T. Monoliths as multiphase reactors: a review. AIChE J. 2004, 50, 2918. (4) Satterfield, C. N.; Ozel, F. Some characteristics of two-phase flow in monolithic catalyst structures. Ind. Eng. Chem. Fundam. 1977, 16, 61. (5) Barnea, D.; Luninski, Y.; Taitel, Y. Flow pattern in horizontal and vertical two phase flow in small diameter pipes. Can. J. Chem. Eng. 1983, 61, 617. (6) Taitel, Y.; Bornea, D.; Dukler, A. E.; Modeling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AIChE J. 1980, 26, 345. (7) Mishima, K.; Hibiki, T. Some characteristics of air-water two-phase flow in small diameter vertical tubes. Int. J. Multiphase Flow 1996, 22, 703. (8) Mishima, K.; Ishill, M. Flow regime transition criteria for upward two-phase flow in vertical tubes. Int. J. Heat Mass Transfer 1984, 27, 723. (9) Coleman, J. W.; Garimella, S. Characterization of two-phase flow patterns in small diameter round and rectangular tubes. Int. J. Heat Mass Transfer 1999, 42, 2869. (10) Triplett, K. A.; Ghiaasiaan, S. M.; Abdel-Khalik, S. I.; Sadowski, D. L. Gas-liquid two-phase flow in microchannels. Part I: two-phase flow patterns. Int. J. Multiphase Flow 1999, 25, 377. (11) Zhao, T. S.; Bi, Q. C. Co-current air-water two-phase flow patterns in vertical triangular microchannels. Int. J. Multiphase Flow 2001, 27, 765. (12) Chen, W. L.; Twu, M. C.; Pan, C. Gas-liquid two-phase flow in micro-channels. Int. J. Multiphase Flow 2002, 28, 1235. (13) 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. 2005, 44, 4884. (14) Chen, L.; Tian, Y. S.; Karayiannis, T. G. The effect of tube diameter on vertical two-phase flow regimes in small tubes. Int. J. Heat Mass Transfer 2006, 49, 4220. (15) Lockhart, R. W.; Martinelli, R. C. Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chem. Eng. Prog. 1949, 45, 39. (16) Chisholm, D. A theoretical basis for the Lockhart-Martinelli correlation for two-phase flow. Int. J. Heat Mass Transfer 1967, 10, 1767. (17) Triplett, K. A.; Ghiaasiaan, S. M.; Abdel-Khalik, S. I.; LeMouel, A.; McCord, B. N. Gas-liquid two-phase flow in microchannels. Part II: Void fraction and pressure drop. Int. J. Multiphase Flow 1999, 25, 395. (18) Zhao, T. S.; Bi, Q. C. Pressure drop characteristics of gas-liquid two-phase flow in vertical miniature triangular channels. Int. J. Heat Mass Transfer 2001, 44, 2523. (19) Kawahara, A.; Chung, P. M. Y.; Kawaji, M. Investigation of twophase flow pattern, void fraction and pressure drop in a microchannel. Int. J. Multiphase Flow 2002, 28, 1411. (20) Dukler, A. E.; Wicks, M., III; Cleveland, R. G. Frictional Pressure drop in two-phase flow. AIChE J. 1964, 10, 38. (21) Lee, H. J.; Lee, S. Y. Pressure drop correlations for two-phase flow within horizontal rectangular channels with small height. Int. J. Multiphase Flow 2001, 27, 783. (22) Yue, J.; Chen, G. W.; Yuan, Q. Pressure drops of single and twophase flows through T-type microchannel mixers. Chem. Eng. J. 2004, 102, 11. (23) Kreutzer, M. T.; Kapteijn, F.; Moulijn, J. A.; Kleijn, C. R.; Heiszwolf, J. J. Inertial and interfacial effects on pressure drop of Taylor flow in capillaries. AIChE J. 2005, 51, 2428. (24) Akbar, M. K.; Ghiaasiaan, S. M. Simulation of Taylor flow in capillaries based on the Volume-of-Fluid technique. Ind. Eng. Chem. Res. 2006, 45, 5396.

Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 255 (25) Krishna, R.; van Baten, J. M. Simulating the motion of gas bubbles in a liquid. Nature 1999, 398, 208. (26) Wallis, G. B. One Dimensional Two-Phase Flow; McGraw-Hill: New York, 1969. (27) Cicchitti. A.; Lombardi, C.; Silverstri, M.; Solddaini, G.; Zavalluilli, R. Two-phase cooling experiments-pressure drop, heat transfer and burnout measurement. Energ. Nucl. (Milan) 1960, 7, 407. (28) Lin, S.; Kwok, C. C. K.; Li, R. Y.; Chen, Z. H.; Chen, Z. Y. Local frictional pressure drop during vaporization for R-12 through capillary tubes. Int. J. Multiphase Flow 1991, 17, 95.

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ReceiVed for reView July 1, 2007 ReVised manuscript receiVed September 10, 2007 Accepted September 20, 2007 IE070901H

Flow Pattern and Pressure Drop of Upward Two-Phase Flow in

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