Reynolds numbers ReL < 350. Visual observations and fast recordings suggest that the onset of ooding atlow ReL (

focused on the partial delivery problem (and not on incipient ooding), where the liquid downowrate is dierent than the liquid feed rate and co-current upow is observed to occur above the li-

52 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181quid entrance section. The ow system in that study, with a pool of liquid (of constant head)above the channel top, is quite dierent from the ow eld studied here. Tests under similar (con-Keywords: Counter-current gasliquid ow; Narrow vertical channel; Flooding; Film characteristics; Wall shear stress

1. Introduction

In recent years there is considerable interest in developing various types of compact devicessuch as two-phase plate heat exchangers and compact reux condensers. In the latter, for example,the narrow ow passages between plates facilitate the onset of ooding that appears to be a lim-iting factor in their operation as it is associated with possible choking of the upward moving va-pors and with a sharp rise of pressure drop. Therefore, it is important to determine the criticalconditions leading to ooding in order to specify conditions of smooth condenser operation. Inad-equate information on these issues exists in the literature, especially at relatively small ReL and forrather short plates. Consequently, the purpose of this work is to study adiabatic gasliquid coun-ter-current ow, focusing on the onset of ooding, in a vertical rectangular channel with a rathernarrow gap, at distances relatively close to liquid entry, thus simulating conditions of interest forthe aforementioned applications.In counter-current gasliquid ow, the onset of ooding is identied by the critical gas ow rate

at which partial liquid ow reversal is observed. Although ooding has been extensively studied inthe past few decades, the amount of work for counter-current ow in vertical rectangular channelsis limited, particularly for a narrow gap between the plates (Banko and Lee, 1986). The followingbrief survey is mainly concerned with counter-current ow inside rectangular channels, although abrief account of similar studies on ow inside vertical tubes is also provided. Lee and Banko(1984) conducted ooding tests at relatively large ReL (ReL > 800) in a rectangular channel forsteamwater counter-current ow (where only one of the wider sides of the channel is wetted);their results clearly indicated the signicant eect of the channel gap and inclination on ooding.Biage et al. (1989) studied experimentally the eect of gas ow on falling lm characteristics forthe case of airwater counter-current ow in a 250 mm wide vertical channel with a relatively largegap (25 mm), at ReL > 1500; these tests showed that the onset of ooding is associated with drop-let entrainment from standing waves. Roy and Jain (1989) measured the local thickness of a waterlm owing down an inclined plane surface with counter-current gas ow, below critical oodingconditions, at various ReL (ranging from 800 to 2000) focusing on the eect of inclination on lmcharacteristics. A study of the dependence of ooding on duct geometry and inclination was pre-sented by Zapke and Kroeger (2000a,b), at ReL > 400, who additionally investigated the eect ofuid properties using several liquid and gas phases. Vlachos et al. (2001) reported ooding datafor ReL > 200 obtained in a vertical rectangular channel with 5 and 10 mm gap in a study of theeect of ReL and channel gap on ooding mechanism.Osakabe and Kawasaki (1989) conducted top ooding experiments in narrow rectangular pas-

sages of high aspect ratio (i.e., 10 100, 5 100 and 2 100 mm2) at ReL > 400. Their work wasstant static head) liquid feeding conditions in vertical rectangular channels are reported by Larson

et al. (1994), Sudo et al. (1991) and Sudo (1996). Larson et al. (1994) studied the eect of locationand geometry of the air injector on counter-current ow development inside very narrow rectan-gular channels of 1.1 and 2.2 mm gap. Sudo et al. (1991) and Sudo (1996) carried out experimentsin channels of various dimensions (with duct gap and width 2.312.3 mm and 3366 mm, respec-tively); they reported that the role of channel gap on ooding is very important, while no signif-icant eect of the corresponding length was observed.Dukler et al. (1984) and Zabaras (1985) studied the structure of the wavy interface on a falling

lm, at counter-current gasliquid ow inside a 2 m long vertical tube, at various locations, forReL higher than 300, while Lacy and Dukler (1994) focused their study on the liquid feed regionfor the same range of ReL. The results of these studies are subsequently compared with the presentdata, wherever is possible. Statistical lm characteristics were obtained by Karimi and Kawaji(2000) in a vertical tube for airkerosene counter-current annular ow at ReL > 1400. Similarmeasurements for airwater annular ow were reported by Vijayan et al. (2001, 2002) who fo-cused their study mainly on post-ooding conditions; the dependence of ooding mechanismon tube diameter was investigated as well. Recently, Mouza et al. (2002) studied ooding atReL > 100 in small diameter tubes (i.d. 6, 7, 8 and 9 mm) with air and two liquids (water andkerosene); the dominant ooding mechanism in almost all cases was found to be wave growthand upward dragging by the gas, initiated at the liquid exit. The strong eect of small tube dia-meter was evident in that study.The above brief literature survey suggests that studies of vertical counter-current gasliquid

ow have been restricted to measurements of the time-averaged lm thickness and of pressuredrop at rather high ReL (ReL > 1500). Other detailed measurements such as instantaneous wallshear stress (not available for the geometry studied here), complementing liquid lm thicknessdata could provide important insight into the problem at hand. An electrochemical techniquewas used by Wragg and Einarsson (1971) and Zabaras (1985) to investigate ow conditionsat the wall in vertical counter-current ow in tubes. Wragg and Einarsson (1971) presented arather large set of time-series of wall shear stress at various liquid and gas ow rates below crit-ical ooding conditions, for ReL between 40 and 1700. Their work, however, mainly dealt withthe qualitative similarity between traces of the wall shear and of the corresponding lm thick-ness. Zabaras (1985) made simultaneous measurements of lm thickness and wall shear stressand investigated the inuence of the imposed gas ow on the wall shear stress at two locationsalong the test tube (at distances 0.15 and 1.7 m from the liquid entry). Several studies of thehydrodynamic structure of rather thick lms (at ReL > 1500), in counter-current annular ow,are also reported for vertical tubes (e.g., Karimi and Kawaji, 1999, 2000), aimed at betterunderstanding the eect of gasliquid interaction on velocity distribution. It will be noted thatin such studies, eorts to measure the velocity prole within the lm were made; however, thewall shear stress was not measured directly but was estimated through the measured velocityprole.In this paper, a description of the experimental equipment and procedures is presented rst.

Visual observations are reported next on the evolution of waves on falling lms, providing aqualitative description of surface restructuring, due to counter-current gas ow, as well asan interpretation of the ooding data. Instantaneous local lm thickness data as well as wallshear stress data and their statistical analysis are reported next. Finally, the data are interpreted

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 53and correlated.

2. Experimental setup and procedures

The experiments were carried out in a specically designed and constructed experimentalfacility (Fig. 1a), the same one used to study developing free falling lm characteristics (Drosos

54 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181Fig. 1. (a) Schematic diagram of experimental ow loop: (1) cross-sectional view of the test section; (2) measurementstations-probes; (3) porous sections; (4) air entrance section; (5) phase separator; (6) liquid storage tank. (b) Detailedcross-sectional view of the air entrance section. (c) Cross-section of the channel.

thatfrom

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 55top of the measuring section there is a phase separator from which the liquid phase is fed backto the storage tank, whereas air is released to the atmosphere. Air and liquid are at ambienttemperature (approx. 20 C).Before each set of experiments with water and butanol solution the inner surface of the Plex-

iglas test section is cleaned by a dilute detergent solution and its wettability is improved by treat-ing it with a silica sol. Such treatment is not necessary when an electrolytic solution is employedfor shear stress measurements. This uid is an aqueous solution of potassium ferri-cyanide(0.01 M) and potassium ferro-cyanide (0.05 M), with sodium hydroxide (1 M) used to suppressionic migration eects. The ferri-cyanide diusivity is obtained as proposed by Gordon et al.(1966). This solution is titrated prior to and after the experiments to determine the exact bulk con-centration of ferri-cyanide ions. During the experiments the solution in the tank is preserved in anitrogen atmosphere to prevent dissolvation of oxygen, whereas the Plexiglas channel and all theother transparent sections of the experimental ow loop are covered by aluminum foil which pro-tects the photo-sensitive electrolytic solution from light exposure. Apart from the electrolytic solu-tion three liquids were also used, i.e., water, 1.5% and 2.5% w/w aqueous n-butanol solutions; liquidphysical properties are listed in Table 1. The dilute butanol solutions were employed to obtainsignicantly smaller surface tension, with practically no eect on the density and viscosity,compared to water. The stability of the butanol solutions was checked prior to and after eachvaries between 10% and 15% of the gas inlet rate. The removed-gas ow rate is subtractedthat of the inlet gas in order to determine the true gas ow rate in the test section. Onet al., 2004). Two-phase ow is developed in a 70 cm high and 12 cm wide rectangular channelmade of Plexiglas to facilitate visual observations. The spacing between the two parallel platesis 10 mm and the active length of the test section is 38 cm. Air enters the device at the bottomthrough a specially made section (i.e., honeycomb) to achieve well-developed conditions beforeit contacts the falling liquid lm. The liquid phase is introduced uniformly near the top of thechannel and removed at the bottom through porous sections, made of stainless steel, with nominalpore size 100 lm (covering the entire active channel width) and embedded ush with the main atsurfaces in order to facilitate smooth liquid entry. Care is taken to uniformly feed the liquid and tominimize external disturbances upon lm formation, by xing a small liquid reservoir right behindeach of the porous segments, which form one of the reservoir walls. The equipment arrangementprovides a suciently long entry section for the gas (Fig. 1b) and minimizes the possibility ofarticially creating disturbances at the liquid exit section.

2.1. Experimental conditions

The liquid and gas ow rates are measured by a bank of calibrated rotameters covering therange of the present experiments (QL = 0.10.7 l/min for the liquid phase and QG = 620 l/sfor the air). In this work, ReL is dened as

ReL 4CLlL1

where CL is the liquid mass ow rate per unit length of the plate and lL is the dynamic viscosity. Asmall amount of gas ows out, together with the liquid lm removed in the lower porous section

56 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181experiment by measuring the surface tension and was found to be satisfactory (with an uncer-tainty of about 5%).

2.2. Film thickness measurements

The lm thickness measurements were taken along the middle of the plate using the well-knownparallel-wire conductance technique described elsewhere (e.g., Paras and Karabelas, 1991).Two probes (described by Drosos et al., 2004) were used, located in the lower half of the channelwhere incipient ooding is observed to occur, at distances from the liquid entry 19 cm and 36 cm.Two additional probes, at 22 cm and 33 cm from liquid entry (i.e., located 3 cm from the mainprobes) were employed to determine the celerity of surface waves.

2.3. Wall shear stress measurements

The local wall shear stress was measured using a well-known electrochemical technique (Reissand Hanratty, 1962). The shear stress measurements were taken along the middle of the plate, atthe same locations where lm thickness was measured (i.e., at x = 19 and 36 cm); double (orsandwich) probes were employed to determine both the magnitude and the direction of the wallshear stress. Each probe consists of two closely spaced parallel platinum strips 0.1 mm thick andapproximately 4 mm wide, arranged normal to the main ow direction and mounted ush withthe inner channel wall. One of the lower porous sections, having a much greater surface area thanthe probe (cathode), acts as the anode. The wall electrodes have to be calibrated in situ in verticalsingle-phase ow, to account mainly for imperfections of their surface. The calibration is carriedout for each electrode prior to and after the main experiments.Data sets of lm thickness and liquid-to-wall shear stress are collected for a period of 80 s

Table 1Physical properties of the liquids used

qL, kg/m3 lL, mPa s r, mN/m

Water 1000 1.00 751.5% butanol solution 995 1.03 502.5% butanol solution 980 1.09 40Electrolytic solution 1080 1.25 42(with a sampling frequency of 125 Hz) and a period of 40 s (with a sampling frequency of250 Hz), respectively.

2.4. Visual observations

Observations of the ow pattern at various locations along the plate were facilitated by fastvideo recordings, using a high speed camera (Redlake Motion Scope PCI). The latter was alignedat two positions, one normal to the main plates and another (parallel to the plates) focusing insidethe gap; a photoood was used as a light source. A diuser was placed in front of this source, toimprove lighting of the main channel, aiming at better image quality of the growing waves.

3. Visual observations

Visualization experiments were carried out in order to study qualitatively the ow pattern dur-ing counter-current ow and to determine conditions associated with the onset of ooding. Eachtest started by rst xing the required liquid ow rate; next the gas ow rate was progressivelyincreased stepwise until the onset of ooding. The latter is dened as the condition where at leastpart of the liquid ow is reversed in direction and carried above the liquid entrance section, even in theform of droplets (Hewitt, 1995).The pictures in Fig. 2 are typical of liquid lm development (near the liquid entrance) at var-

ious gas ow rates, including incipient ooding conditions. It is noted that these characteristicsare qualitatively nearly the same for the three liquids employed. Film surface restructuring atthis locationdue to the gasliquid interactionseems to be more pronounced, even at thelower gas velocities examined in the present study, compared to that at the lower section ofthe channel (shown in Fig. 3). At small supercial gas velocity UGS, i.e., far below critical con-ditions, there is an entrance length where the lm surface appears to be smooth and undis-turbed. Downstream nearly two-dimensional disturbances develop rst, evolving into a three-dimensional, small-amplitude wave structure (e.g., Pierson and Whitaker, 1977; Chang, 1994;Drosos et al., 2004). By increasing the gas ow rate, the smooth entrance length is reducedand the initial small two-dimensional waves that appear near liquid entrance evolve rapidly intothree-dimensional ones; furthermore, the lm between the waves appears to be covered by smal-ler waves (ripples), having practically no undisturbed regions. As ooding is approached, thewavy disturbances on the lm surface move upstream, covering the entire plate surface upto the liquid entry; additionally, the waves lose any feature of two-dimensionality. The eectof liquid ow rate is noticeable, if one compares pictures taken at ReL = 130 and 225(Fig. 2). At the entrance section, lower ReL are associated with nearly two-dimensional wavesover a larger range of gas ow rates (e.g., Fig. 2a, UGS = 6.5 m/s). It should be noted that forthe range of ReL values and for all gas ow rates examined in this study (UGS < 11 m/s) noliquid reversal is observed (by naked eye or detected in the recordings) to occur at this uppersection of the channel. However, incipient ooding is observed in the lower half of the channel,mostly close to the liquid exit area, as shown in Fig. 3 where the inuence of gas supercialvelocity on wave structure is shown for the section between 22 and 32 cm from liquid entry.It appears that for low gas ow rates the wave structure at the lower half of the channel re-mains qualitatively nearly the same, close to the one corresponding to the free falling lm case,as it has been also reported elsewhere for the case of counter-current ow in tubes far below theliquid entry (e.g., Mouza et al., 2002).For conditions near the onset of ooding, notable features of the rather strong gasliquid inter-

action are the fairly coherent large waves (like the one depicted in Fig. 3b for UGS = 7.5 m/s,t = 0 ms) which seem to be momentarily arrested near the liquid exit by the counter-currentlyowing gas. Such a wave, when formed, apparently cannot be transported (as a whole) upwardsover a signicant distance, but displays a tendency to become unstable and to eventually break upinto droplets that follow the gas up-ow. It seems that most of these droplets are deposited on thesurface of the falling lm; only a small part appears to pass above the liquid injection station.After that event, the total liquid ow downwards is restored until a new wave is arrested near

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 57the exit and the process is repeated. The evolution of similar unstable, laterally coherent waves

58 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181which appear intermittently near the exit has been also reported in the literature (e.g., Zabaras,1985; Lacy and Dukler, 1994; Vijayan et al., 2001) for large ReL (ReL > 1000) and large diametertubes (D > 20 cm); for the case of a rectangular channel with a rather large gap (25 mm), Biageet al. (1989) observed that some large waves did start moving upwards, but (almost immediately)fell downward and vanished. On the contrary Mouza et al. (2002) observed that the small tubediameter facilitates the development of (circumferentially) coherent waves which tend to grow andbe dragged upward by the gas, at incipient ooding; the latter takes place at signicantly smaller

Fig. 2. Eect of gas velocity on liquid lm ow on the channel main plate, at the entrance region (x 0 to 12 cm), for2.5% butanol solution at two Reynolds numbers: (a) ReL = 130 and (b) ReL = 225.

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 59velocities than those observed in large size tubes, for the same range of ReL ( 250, it is clear that unstable waves (beforebreaking-up) tend to reach the wavy lm on the opposite plate and to cause partial blockageof the gap (local bridging), due to the rapid increase of their amplitude. The liquid of these

Fig. 3. Liquid lm ow near the end of the channel for 2.5% butanol solution at ReL = 260. (a) Flow pattern at variousgas velocities. (b) Formation and disintegration of a coherent wave at ooding, over a period of 328 ms.

colliding waves is forced by the gas to move upwards in the form of a slug, over some distance(less than about 10 cm) above the point where bridging is initially observed, until the gas eventuallypenetrates this slug leading to its disintegration. The largest portion of the liquid is then trans-ported upwards (beyond the liquid inlet) in the form of rather large drops, while the remainingliquid seems to be dispersed in the lateral direction.

4. Critical ooding velocities

An overview of critical ooding conditions is provided in Fig. 4, for the three liquids tested,obtained by means of visual observations and fast video recordings. The data are plotted in termsof supercial gas velocity, UGS QGwb (QG, w and b are the volumetric gas ow rate, the channelwidth and gap respectively), at incipient ooding, vs. the corresponding ReL; three main regionscan be readily observed:

Region A. This is the region of fairly strong dependence of critical ooding velocity on ReL,where the supercial gas velocity is inversely proportional to Re . This trend is the usual

6 0.05WB1

Water1.5% Butanol

60 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 51813 0.000 50 100 150 200 250 300 350 400

B2

Re L

h mean2.5% Butanol

Fig. 4. Supercial gas velocity, UGS, vs. ReL (for the three liquids examined) at the onset of ooding, and mean waveL

one referred to in previously published studies (e.g., Hewitt, 1995; Banko and Lee, 1986). Region B. By further increasing the liquid ow rate in the range ReL 150250, the oodingvelocity tends to be weakly dependent on ReL. Free falling lm data for all three liquids exam-ined (Drosos et al., 2004) may be used for a simple qualitative interpretation of this trend; thesedata show that the mean wave amplitude above the substrate, Dhmean, at ReL 150250 isroughly constant (as shown in Fig. 4 for the case of 2.5% butanol solution) which is in accordwith the same trend of the ooding velocity.

Region C. For ReL > 250 a somewhat stronger dependence of critical velocity on ReL is againevident. A plausible qualitative explanation may be based on the visual observations and the

9

12

15

0.10

0.15

0.20

hm

ean

, mm

UG

S, m

/s

A B Camplitude Dhmean for a free falling lm of 2.5% butanol solution vs. ReL at x = 36 cm (Drosos et al., 2004).

hydrodynamic analysis for the free falling lm case (Drosos et al., 2004) indicating a moreunstable ow with increasing ReL; thus, the onset of ooding may be facilitated. This is con-sistent with the observed dominant mechanism for incipient ooding at ReL > 250, which ischaracterized by wave growth and local bridging. Furthermore, it is noted (Drosos et al.,2004) that the mean lm thickness (in the absence of gas) continues to increase with ReL in thatrange, even though the wave amplitude attains a nearly constant value. Therefore, a reducedfree cross-sectional area is available for gas ow at ReL > 250, requiring a smaller gas velocityfor the initiation of ow reversal of the unstable liquid lm.

The rather strong inuence of the surface tension on ooding is evident in Fig. 4, the butanolsolutions displaying lower critical velocities than water. This is a well-known trend (e.g., Bankoand Lee, 1986; Zapke and Kroeger, 2000a,b; Mouza et al., 2002), attributed to the lower surfacetension of butanol solutions which is associated with increased wave instability, as reported in theliterature (e.g., Alekseenko et al., 1994), and already discussed in connection with free falling lmdata taken by Drosos et al. (2004) in the same channel. The inuence of surface tension on wavegrowth and ooding is also shown and discussed in subsequently presented Figs. 8 and 9.In Fig. 5 a comparison is shown of the present data for water with some published data for

counter-current ow inside rectangular channels in terms of dimensionless velocities U L, UG.

The channel gap, b, is adopted as the representative length scale for all data to facilitate compar-ison. The experimental results obtained by Mishima (1984), correspond to a rectangular channel

2.4

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 610.3

0.6

0.9

1.2

1.5

1.8

2.1

0 0 .075 0.15 0.225 0.3

b= 5.3b= 12.3b= 5.3b= 12.3Mishima, 19845 mm10 mm

(UL* ) 0.5

(UG*

)0.5

w =33 mm

w =66 mm

(Water)

Sudo et al., 19 91(Water)

K1

K2

K3

Vlachos et al., 2001 (Water)

Eqn. [2]Sudo et al. (1991) Zapke & Kroger (2000)

Water (present study)

Fig. 5. Comparison of the present ooding data for water with literature experimental data, for rectangular channels,

and with proposed empirical correlations.

havinand

gas velocity tends to increase quite sharply, whereas for the case of circular tubes at relatively highReLWall

ratioOn

lineawateas L

Th(mar

62 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181U G 0:0742

U LOh0:15L

6

where OhL is the Ohnesorge number dened as

OhL l2Lq br

s7e empirical correlation proposed by Zapke and Kroeger (2000a,b) is also included in Fig. 5ked as Line K3):C2 0:66 bw 0:25

4

Here Bo* is a modied Bond number dened as

Bo wbqL qGgr

5

where r is the liquid surface tension.C1 0:5 0:0015Bo1:3 3The values of parameters C1 (=0.708) and C2 (=1.136) are greater than those proposed in the lit-erature (e.g., Hewitt, 1995) but consistent with the trend reported by Sudo et al. (1991) showingthat C1 increases with both the width and the gap of the channel while C2 increases with increasingwidth.Fig. 5 also shows the estimated ooding velocities for conditions of the present study using

again a Wallis-type expression (marked as Line K2 in Fig. 5) for which the values of parametersC1, C2 are obtained from the empirical correlation proposed by Sudo et al. (1991):r dependence is observed between UL and UG. Such a correlation for the present data (forr), at the high liquid ow rates examined, is described by the following expression (markedine K1 in Fig. 5):

U Gp 0:708 U Lp 1:136 2for data correlation would be appropriate only at relatively large liquid ow rates, for which a of channel width over its gap, w/b, decreases.the basis of their ooding velocities, Sudo et al. (1991) suggest that a Wallis-type expressionU L ULS

qLgbqL qG

rand U G UGS

qG

gbqL qGr

Sudo et al. (1991) report that the aforementioned sharp increase is more pronounced when thea linear increase is reported (e.g., Banko and Lee, 1986; Hewitt, 1995). The well-knownis (1969) correlation also involves a linear relation between:rectangular channels) reveal that, at very low liquid ow rates, with decreasing UL the criticalg a width and a gap of 40 mm and 5 mm, respectively; these data are presented by OsakabeKawasaki (1989). Experimental results (of the present study and published ones in narrowL

a0.4

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 63To obtain Line K3 in Fig. 5, the Ohnesorge number was computed with water properties, to becompatible with the rest of the data.These correlations deviate signicantly from the present data, probably because the one pro-

posed by Sudo et al. (1991) was obtained with smaller rectangular channels (i.e., with much great-er values of U L, between 0.25 and 1.5), while in Zapke and Kroeger (2000a,b) higher liquid owrates where used, although ooding tests were conducted at channels of comparable dimensionswith the present one.

FrLS

Fig. 6. Correlation of ooding conditions in dimensionless form, (FrGS/Ka0.4) vs. FrLS, for the three liquids examined.Inby usKa, a

wher

It isJame

5. In

5.1. F

Ththe lo0 0.001 0.00 2 0.003

0.024

0.04F

r GS

/K

K0.06

0.08Water1.5% Butanol2.5% ButanolEqn. [8]Fig. 6 an attempt is presented to relate the ooding data with the liquid physical propertiesing the dimensionless gas Froude number, FrGS, (which is equal to U

2G ) and liquid Kapitza,

nd Froude, FrLS, numbers. The data are correlated by the following expression:

FrGS cflKa0:4Fr0:15LS 8e the constant c = 0.0138 is used, and

FrGS qGU2GS

gbqL qG; FrLS qLU

2LS

gbqL qG; Ka r

lL

qLlLg

1=3noteworthy that the critical ooding velocity is proportional to Ka0.2 as Cetinbudaklar andson (1969) obtained in their theoretical stability analysis.

terpretation of instantaneous lm thickness data

ilm thickness characteristics

e study of the variation of lm thickness variation with supercial gas velocity is focused onwer half of the channel where initiation of ooding mainly occurs. Film thickness traces, not

presented here, clearly conrm the preceding qualitative visual observations. In Fig. 7 the varia-tion of lm thickness with supercial gas velocity is presented at x = 19 cm and x = 36 cm, forwater and 2.5% butanol solution at various ReL. Shaded areas in this and in subsequent graphsindicate approximately post-ooding conditions. It is evident that an increasing gas rate results inincreasing mean thickness at both locations; this increase is rather small at low UGS, but as the gasrate approaches the critical value for ooding, the mean thickness increases sharply. However, atx = 19 cm this trend appears to be less sharp, whereas at x = 36 cm the lm thickness reaches amaximum value associated with the initiation of ooding which is mainly observed close to thatlocation. Indeed, this maximum corresponds to the critical gas velocity obtained from visualobservations (Fig. 4). Similar observations for the increase of lm thickness with supercial gasvelocity (but of smaller magnitude) are reported in the literature (Zabaras, 1985; Lacy and Duk-ler, 1994) for ReL values greater than those of the present study. At large ReL (ReL > 2000), Biageet al. (1989), Karimi and Kawaji (2000) and Vijayan et al. (2002) report that the mean lm

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

110140175210245280310

UGS, m/s UGS, m/s

x=19 cm

0 2 4 6 8 10 12

110140175210245280310

x=36 cm

(a)

h mea

n,m

m

x=19 cm x=36 cm

64 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181UGS, m/sUGS, m/s

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

100130165195225260290

h mea

n,m

m

0 2 4 6 8 10 12

100130165195225260290

(b)

Fig. 7. Mean lm thickness vs. UGS for various ReL at two locations along the test plate (x = 19 and 36 cm) for

(a) water and (b) 2.5% butanol solution.

thickness displays a tendency of remaining constant, being almost independent of gas ow rate,up to the ooding point where a sharp increase is observed; a similar observation is reported byZabaras (1985) for ReL > 1500. It must be noted that (in the present tests) at the onset of oodingthe measured maximum lm thickness at x = 36 cm for ReL < 250 is signicantly lower comparedto the length of the probe at that location (2.7 mm), whereas for ReL > 250, it is very close to thislength. Therefore, the true mean lm thickness at incipient ooding for ReL > 250 may be largerthan that measured with the present probes; the same is apparently true for the butanol solution(Fig. 7b).Another parameter of the liquid lm worth examining is the mean wave amplitude above the

substrate, Dhmean, and its variation with UGS. This quantity is the mean value of the dierencebetween a minimum in the lm trace and the preceding maximum, corresponding to a large wave,for all large waves detected in a lm thickness record (Drosos et al., 2004); the procedure for com-puting Dhmean is reported elsewhere (Paras and Karabelas, 1991; Drosos et al., 2004). Fig. 8 shows

0.

0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

110140175210245280310

h mea

n, m

m

UGS , m/s UGS , m/s

x=19 cm

0 2 4 6 8 10 12

110140175210245280310

x=36 cm

(a)

x=19 cm x=36 cm

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 65h m

ean

, mm

UGS , m/sUGS , m/s

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

100130165195225260290

0 2 4 6 8 10 12

100130165195225260290

(b)

Fig. 8. Mean wave amplitude vs. UGS for various ReL at two locations along the test plate (x = 19 and 36 cm) for

(a) water and (b) 2.5% butanol solution.

that the wave amplitude displays a trend similar to that of mean thickness, although in the lattercase the increase near ooding is sharper. These trends for mean lm thickness and wave ampli-tude suggest that a substrate thickening occurs with increasing UGS; this was conrmed by deter-mining the substrate thickness (not presented here due to space limitations).The variation of RMS values of lm thickness under counter-current ow, hcc,RMS, plotted in

Fig. 9 as a dimensionless RMS ratio (by dividing with the corresponding RMS values of lmthickness at zero gas ow, h,RMS) provides additional evidence of the change of wave structure;i.e., the roughness of the liquid lm tends to grow as the gas ow rate increases. Near oodingthere is a very sharp increase of hRMS which is even sharper than that of mean thickness(Fig. 7). By further increasing the gas ow rate beyond critical conditions, it appears that hcc,RMS,hmean, and Dhmean tend to decrease possibly due to the decreasing rate of liquid downow, afterpartial liquid ow reversal.

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 2 4 6 8 10 12

110140175210245280310

UGS, m/s

x=19 cm

h cc,

RM

S/ h

ff,R

MS

0 2 4 6 8 10 12

110140175210245280310

x=36 cm

(a)

6.0

7.0100130165

x=19 cm

100130165

x=36 cm

UGS, m/s

66 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181h cc,

RM

S/ h

ff,R

MS

1.0

2.0

3.0

4.0

5.0

0 2 4 6 8 10 12

195225260290

0 2 4 6 8 10 12

195225260290

(b) UGS, m/s UGS, m/s

Fig. 9. Ratio of RMS values of lm thickness at counter-current ow, over the corresponding RMS values at zero gasow vs. UGS for various ReL at two locations along the test plate (x = 19 and 36 cm) for (a) water and (b) 2.5% butanol

solution.

quency which at high UGS lies roughly between 1 and 3 Hz. Such a reduction of the modal fre-quency is also reported by Zabaras (1985) and Biage et al. (1989); in their studies a dominant

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 67frequency of approx. 25 Hz was obtained at the onset of ooding. Another characteristic featureof the spectral density functions (Fig. 10) is a secondary frequency of approx. 35 Hz that seemsto emerge at both measuring stations and may be attributed to a distribution of wave energy tosmaller waves (possibly to forerunner waves located in front of larger ones) and their signicantgrowth by further increasing gas ow rate. This secondary frequency eventually vanishes as ood-ing is approached. At rather high gas ow rates, an additional pronounced peak of approx. 10 Hzis observed in the spectrum corresponding to x = 19 cm for all liquids examined, especially athigher ReL. This frequency may be indicative of a more gradual restructuring of the ow pat-tern at this location, with increasing UGS, compared to the drastic changes that occur in the lmow near the liquid exit region; the latter are reected in the rather sharp reduction of the modalfrequency at x = 36 cm.By further increasing gas ow rate above critical conditions, the dominant frequency remains

nearly the same, but a decrease of the energy conveyed by the waves is evident possibly due to theInteresting observations can be made by comparing the variation of liquid lm characteristics(with UGS) for the case of butanol solution to those for water (Figs. 79). For the lower surfacetension solution it is evident that the critical gas velocity corresponds to the maximum value ofmean lm thickness at both locations; however this is not the case for water at x = 19 cm. Thistrend is attributed to the fact that the gasliquid interaction results in a more disturbed interfacefor the butanol solution; this disturbance in the liquid lm surface seems to be uniformlyextending upwards, up to the liquid entry. A similar observation is reported by Zabaras (1985)for a low surface tension electrochemical solution used in his study. For water, the present datasuggest that the lm thickness at x = 19 cm continues to increase for conditions beyond ooding(Fig. 7a). The aforementioned dierence between water and butanol solutions (due to dierentphysical properties) also manifests itself in the larger values of hRMS observed for the butanolsolution at both locations and in the steeper increase of RMS values near the ooding pointcompared to those of water.

5.2. Power spectral density of lm thickness uctuations

Typical power spectral densities of lm thickness for water and 2.5% butanol solution areshown in Fig. 10 for conditions near and at the ooding point (including the free falling lm case)for the two locations along the channel, where the inuence of the imposed gas ow on wavestructure is reected.For the free falling lm a pronounced maximum is observed at frequencies between 17 and

20 Hz at x = 19 cm, while at x = 36 cm this maximum corresponds to a frequency between 7and 10 Hz; these frequencies are related to the periodic passage of large waves. For the case oflow gas ow rates (UGS < 5 m/s), it appears that the dominant wave frequency does not changesignicantly compared to the free falling lm case; however, an increase of the energy conveyedby the waves is observed (especially for 2.5% butanol solution). This increase is more pronouncedat higher gas ow rates towards ooding, along with the apparent reduction of the dominant fre-decreasing rate of downow.

00.2

0.4

0.6

0.8

0.01 0.1 1 10 100

04.35.46.5

f, Hz

PS

D

ReL = 100

UGS

(m /s)

.0

0

2

4

6

8

0.01 0.1 1 10 100

7.57.88.69.7

f, Hz

ReL = 100 U

GS

(m/s)

(a)0.0

0.2

0.4

0.6

0.8

0.01 0.1 1 10 100

03.24.36.5

f, Hz

PS

D

ReL = 260

secondaryfrequencyof ~3-5 Hz

UGS(m/s)

.0

0

2

4

6

8

0.01 0.1 1 10 100

6.87.58.69.1

f, Hz

ReL = 260

secondaryfrequencyof ~10 Hz

UGS(m/s)

0

0.2

0.4

0.6

0.8

0.01 0.1 1 10 100

03.64.76.5

f, Hz

PS

D

ReL = 130

secondaryfrequencyof ~3-5 Hz

UGS(m /s)

.0

0

2

4

6

8

0.01 0.1 1 10 100

6.87.58.69.7

f, Hz

ReL = 130

UGS(m/s)

(b)0

0.2

0.4

0.6

0.8

0.01 0.1 1 10 100

03.24.75.8

f, Hz

PS

D

ReL = 290

UGS(m /s)

.0

0

2

4

6

8

0.01 0.1 1 10 100

6.57.58.69.7

f, Hz

ReL = 290 UGS

(m/s)

Fig. 10. Typical power spectral densities for 2.5% butanol solution for various ReL and various UGS, at two locationsalong the test plate: (a) x = 19 cm and (b) x = 36 cm.

68 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10

100130165195225260290

c, m

/s

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10

100130165195225260290

c, m

/s

(b)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 2 4 6 8 10

130195260

c/ cc

UGS , m/s(c)

UGS , m/s

UGS , m/s

Fig. 11. Variation of wave celerity with UGS for 2.5% butanol solution, for various ReL, at two locations along the testplate: (a) x = 19 cm, (b) x = 36 cm, (c) x = 19 cm; wave celerity normalized with mean lm velocity, c/huicc.

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 69

70 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 51816. Interpretation of instantaneous wall shear stress data

6.1. Mean and RMS wall shear stress

The purpose of the measurements was to determine both the direction and the magnitude of thewall stress. The ow direction of the falling lm near the wall was found to be downward at all gasrates examined; i.e., changes of the wall shear stress sign did not appear to occur.Typical shear stress traces at x = 19 cm and 36 cm are depicted in Fig. 12 providing an indica-

tion of the surface structure inuence on wall shear stress uctuations. The sharp peaks in bothcases may be attributed to the gasliquid interface waves which seem to aect local velocity andshear stress at the channel wall. A notable feature of these time-series is the increase of the ampli-tude of shear stress uctuations with gas velocity at both locations; however, it seems that the owpattern in the proximity to the wall is inuenced less by surface waves at location x = 19 cm (espe-cially at low gas ow rates) than at x = 36 cm; i.e., near the liquid exit. Apparently, waves ofsomewhat smaller amplitude (at x = 19 cm) have reduced inuence on local velocities at the wall.A similar trend is also reported in previous studies (Zabaras, 1985; Karimi and Kawaji, 1999;Moran et al., 2002), where it is shown that the substrate lm surrounding a wave structure isstrongly inuenced mainly by the presence of large-amplitude waves, whereas in the region ofdeveloping ow the growing waves cannot exert their inuence down to the wall.The time series of wall shear stress data are statistically analyzed to obtain the time-averaged

wall shear stress, sw,mean, and the corresponding root mean square, sw,RMS. The variation of5.3. Wave celerity

Although a signicant reduction of liquid wave celerity might have been expected on the basisof lm ow retardation due to the interfacial shear stress, the present experimental results showthat (at a given location) there is generally a rather modest reduction with increasing UGS, even athigh gas ow rates, whereas a somewhat stronger dependence on ReL is observed (Fig. 11a, b).The rather small reduction of celerity with increasing UGS may be related to the increase of thewave amplitude, in that the greater interfacial forces may be roughly counterbalanced by theincreasing gravitational eect. Fig. 11c shows the normalized celerity c/huicc data at x = 19 cm,where the mean lm velocity huicc is estimated on the basis of Eq. (A.8) using the measured thick-ness hmean. It is interesting that wave celerity is much greater than huicc well below ooding. Thec/huicc data at UGS = 0 display the trend already discussed by Drosos et al. (2004).It is worth mentioning that near ooding, a chaotic ow pattern prevails, resulting in a poor

correlation; therefore, it is not possible to obtain a reliable estimate of wave celerity through across-correlation function at such high gas velocities. A nearly constant wave celerity is also re-ported by Zabaras (1985) and Biage et al. (1989); in these studies it is noticed that the large waveswhich are correlated well enough for ReL > 700 (even at the onset of ooding) continue to traveldownwards without being decelerated. In the study by Karimi and Kawaji (2000) at high ReL(>2000) a signicant reduction of surface velocity was noticed only for gas ow rates beyondooding.

01

2

3

4

5

0 0.2 0.4 0.6 0.8 1

03.28.6

ww

,N/m

2

t, sec

ReL =135

UGS(m/s)

.0

(a)0 0.2 0.4 0.6 0.8 1

02.27.8

0

1

2

3

4

5

t, sec

ReL =265

UGS(m/s)

.0

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

03.28.6

t, sec

ReL =135

UGS(m/s)

.0

(b)0 0.2 0.4 0.6 0.8 1

03.27.8

0

1

,N

/m2

w

,N/m

2

w,N

/m2

2

3

4

5

t, sec

ReL =265

UGS(m/s)

w

.0

Fig. 12. Typical traces of wall shear stress taken simultaneously for two ReL (ReL = 135 and 265) and various UGS, attwo locations along the test plate: (a) x = 19 cm and (b) x = 36 cm.

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 71

12

3

4

135170200

w

,mea

n, N

/m2

72 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181sw,mean with supercial gas velocity, for various ReL, is presented in Fig. 13. Increasing gas owrate results in decreasing wall stress since the gravity force acting on the liquid is partially counter-balanced by the gasliquid interfacial shear as well as by wall shear forces. By further increasinggas velocity above 67 m/s the ooding point is approached, and a sharp reduction of the meanshear stress is observed. This trend is quite similar to the sharp increase displayed by the mean lmthickness and mean wave amplitude, indicating that the inuence of interfacial gasliquid shearextends down to the solid wall. Additional support of this observation is provided by the RMSvalues of shear stress (Fig. 14) which display a tendency to increase with increasing gas velocity,as also noticed by Zabaras (1985) and Karimi and Kawaji (1999). The RMS values reach a max-imum at the ooding point; this trend is apparently related to the more disturbed surface structureobserved at higher gas ow rates, reected also in the RMS values of lm thickness (Fig. 9).

00 2 4 6 8 10 12

230265295

w

,mea

n, N

/m2

UGS , m/s

UGS, m/s

(a)

0

1

2

3

4

0 2 4 6 8 10 12

135170200230265295

(b)

Fig. 13. Mean wall shear stress sw,mean vs. UGS for various ReL at two locations along the test plate: (a) x = 19 cm and(b) x = 36 cm.

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 730.2

0.4

0.6

0.8

1.0135170200230265295

w,R

MS

, N/m

26.2. Power spectral density of wall shear stress uctuations

The inuence of the interfacial wave structure on the ow pattern near the wall is also examinedvia typical power spectral density functions of wall shear stress uctuations, depicted in Fig. 15 forthree dierent values of ReL, corresponding to a location near the liquid exit (x = 36 cm) wherethe surface structure seems to be better developed than at the upper or at the middle section ofthe channel. A distinct peak, at frequencies between 7 and 10 Hz, is evident for the free falling lmcase. This dominant frequency seems to be independent of liquid ow rate (ReL), in contrast to theobserved reduction of its value when gas ow rate increases; at conditions near ooding the modalfrequency reaches a value between 2 and 4 Hz. A similar trend is reported by Wragg and Einars-son (1971) and Zabaras (1985). Another notable feature of the spectral density functions in Fig. 15

0.00 2 4 6 8 10 12

w,R

MS

, N/m

2

UGS , m/s

UGS , m/s

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

135170200230265295

(b)

Fig. 14. RMS values of mean wall stress vs. UGS for various ReL at two locations along the test plate: (a) x = 19 cm and(b) x = 36 cm.

02

4

6

8

10

0.1 1 10 100

03.25.47.59.7

PS

D

f, Hz

ReL =135UGS

(m/s)

.0

0

2

4

6

8

10

0.1 1 10 100

03.25.47.58.6

PS

D

f, Hz

ReL=200UGS

(m/s)

.0

0

2

4

6

8

10

0.1 1 10 100

03.25.46.87.8

PS

D

f, Hz

ReL=265UGS

(m/s)

.0

Fig. 15. Typical power spectral densities of wall shear stress uctuations at counter-current ow for various ReL atx = 36 cm.

74 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181

is the apparent increase of the maximum (peak) value with increasing gas ow rate. An additionalfrequency peak at approx. 35 Hz seems to emerge as gas ow rate increases (especially at higherReL) and may be attributed to the signicant growth of the smaller waves that inuence the oweld close to the wall.It is clear that the spectral density functions of shear stress and liquid lm thickness are very

similar indicating the inuence of waves on lm ow. It should be pointed out, however, that thissimilarity does not imply that instantaneous shear stress is in-phase with lm thickness; the pres-ent data, not taken simultaneously, cannot resolve this issue. According to Zabaras (1985), wheresimultaneous measurements of lm thickness and shear stress were made, the correlation betweenthese two quantities improves as the distance of the measuring station, below liquid feed, in-creases, whereas it is very poor at a region close to that of wave inception. For the latter regiona better correlation is established only at rather high gas ow rates.

6.3. Film thicknesswall shear stress interrelationship

The present experimental results reveal that even at rather high gas velocities, just below theooding point, the lm ow is dominated by the downward wave motion. Furthermore, it appearsthat there is a rather insignicant net eect of imposed gas ow on wave celerity up to aboutooding conditions; indeed it seems that wave celerities are nearly equal to the ones corresponding

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 75to zero gas ow for the same liquid rate (Fig. 11). However, the eect of the interfacial shear onlm ow is not negligible; counter-current gas ow strongly aects all the other wave character-istics and it is associated with drastic liquid lm restructuring.In Appendix A, a simple model is presented to estimate the average lm thickness at counter-

current ow, hcc, and the mean wall shear stress, sw,mean, at UGS below the onset of ooding. For

1.0

1.2

1.4

1.6

1.8

2.0

0 1 2 3 4 5 6 7

100130165195225260290

194289h c

c / h f

f

UGS , m/s

(100)(195)(290)

Eqn. [9]

Fig. 16. Comparison of the present experimental results plotted as hcc/h vs. UGS for 2.5% butanol solution at x = 36 cmwith predictions based on Eq. (9). Predictions correspond to ReL values of 100, 195 and 290.

steadtities

0.6 170200 c

c/qLg hff 2 qLghff hff

sw;mean lLdUx

qLg qGg 2 sG 2 si

hcc si 10FirentUGSsimilwithtakenexpe(9) acurre

7. Co

Itshortgrowin th(>25y, one-dimensional lm ow with imposed shear, si, at the gasliquid interface, these quan-may be calculated from the following expressions:

qL qGg 2w sG 2arb si hcc 3

3 si hcc 2

1 0 90.0

0.2

0.4

0 1 2 3 4 5 6 7

230265295135200295

UGS , m/s

(135)(200)(295)

Eqn. [10]

Fig. 17. Comparison of the present experimental results plotted as scc/s vs. UGS for the electrolytic solution atx = 36 cm with predictions based on Eq. (10). Predictions correspond to ReL values of 135, 200 and 295.0.8

1.0

1.2

135ff

76 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181dy y0 w arb

g. 16 displays the predicted variation of the ratio of the mean lm thickness at counter-cur-ow over the mean thickness for the free falling lm case, hcc/h, with supercial gas velocityfor the case of 2.5% butanol solution; the experimental results are also included. In Fig. 17 aar comparison is shown of the estimated wall shear stress at counter-current ow normalizedwall stress for the laminar ow case (at zero gas ow), scc/s, with the experimental resultsat x = 36 cm. In both comparisons, fair agreement is obtained between predictions and

rimental data with a maximum uncertainty 15% at the higher UGS examined. Thus, Eqs.nd (10) may be used for the prediction of mean lm thickness and wall shear at counter-nt ow for given gas and liquid ow rates.

ncluding remarks

is observed that incipient ooding takes place mainly close to the liquid exit of the rathertest-section employed. The onset of ooding at ReL smaller than 250 is associated withth, ow reversal (momentarily) and disintegration of waves covering part of the lm surfacee lateral direction; liquid droplets torn-o those waves are entrained by the gas. At higher ReL0), local bridging of surface waves is the dominant mechanism. The wave structure of the

free flutio

lm thickness which tend to increase signicantly even at gas ow rates well below the oodingpoint. Flooding transition corresponds to a distinct change in the form of the power spectrum

attributed to the increased gasliquid interfacial shear caused by the increase of the mean waveamplitude. The increasing RMS values of wall shear stress uctuations, with increasing gas rate,

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 77are in accord with a similar increase displayed by the RMS of instantaneous lm thickness and isindicative of the inuence of interfacial waves on ow close to the wall. At relatively low gasvelocities (UGS < 5 m/s) this wave eect is found to be more pronounced near the liquid exit(i.e., at x = 36 cm), whereas ow at the wall is almost unaected by waves at locations closerto the liquid entry (i.e., at x = 19 cm). Additional evidence on the inuence of large waves onliquid ow at the wall is provided by the similarities between the lm thickness and wallshear power spectra.In general, the falling lm characteristics obtained in this study show that signicant changes

occur in liquid lm ow, even close to the channel wall, under conditions approaching ooding.The proposed simple model for predicting the mean wall shear stress and mean lm thickness (be-low the onset of ooding) provides only rst estimates. To complement the present data and de-velop reliable predictive tools for ooding, additional work is required on ow in narrowpassages. The eects of gap size/shape and of liquid properties need special attention.

Acknowledgement

The authors wish to thank Mr. A. Lekkas, Mr. T. Tsilipiras and Mr. F. Lambropoulos for thetechnical support.

Appendix A. A model to estimate mean lm thickness and mean wall shear stress in counter-currentgasliquid ow

Assuming steady, laminar one-dimensional ow subject to interfacial shear one obtains:

1 dp

g lL d2Ux

2 0 A:1and it is associated with a signicant reduction of the dominant wave frequency. Furthermore,the energy conveyed by the waves is increasing with increasing ReL or UGS.The wall shear stress measurements show no evidence of liquid lm ow reversal up to the onset

of ooding. The observed reduction of mean wall shear with increasing gas ow rate may beInstantaneous lm thickness and wall shear stress measurements were made at two locations atthe lower half of the channel. By increasing the gas ow rate a relatively small increase of meanlm thickness, hmean, and of mean wave amplitude, Dhmean, is observed well below the criticalooding velocity, compared to the rather sharp increase of these quantities near the onset ofooding. The eect of gas velocity appears to be comparatively stronger on the RMS values ofalling lm, on which gas shearing is applied, tends to inuence to a great extent wave evo-n (at increasing UGS) and ooding.qL dx qL dy

with

TheaxialIn

For

wher G G i

x

r i

while

In th

78 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 51813lLwhere absence of gas ow, Eq. (A.8) is reduced to

Q qLgh3ff A:9Q Z hcc0

Uxydy qLg qGg 2w sG 2arb si

3lLh3cc

si2lL

h2cc A:8Uxy lLhccy

2lL

y A:7

the liquid ow rate, per unit width of the channel, may be obtained asThe velocity distribution is obtained by integrating Eq. (A.6):

qLg qGg 2w sG 2a b si

y2

ssyx lL dy qLg qGg w sG arb si hcc y si A:6Substitution of Eq. (A.4) in Eq. (A.2) leads to an expression for the shear stress distribution:

dU 2 2 ar b 2hccb A:5where ar is the void fraction in a given section of the channel: dpdx

qGg 2

wsG 2arb si 0 A:4and Eq. (A.3) leads to dpdx

AG qGgAG sGSG siSi 0 A:3

e A , S and S are dened in Fig. 1c. The uctuation at the gasliquid interface is neglectedconsidered to be wetted by the falling lm, a force balance in the gas core gives:syx lL dy qLg dx hcc y si A:2

counter-current ow in a vertical rectangular channel in which only the wide plates aredUx dp origin of coordinates is placed at the channel wall; y is the distance from the wall and x is thecoordinate.tegration of (A.1) yields:dUxdyyhcc

silL at y hccboundary conditions

Ux 0 at y 0e h is the mean thickness for the case of the free falling lm.

Giconsi

1

whicinter

are d

hN

E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 79C3 41:3BoC40:25e 109:07=Boe A:17C4 1:63 4:74 A:18ened as follows. Here, the value of hN (obtained from the well-known Nusselt formula;3CLlLq2Lg

1=3 3

4

l2L

q2Lg

1=3Re1=3L ) is used for convenience, instead of h.where De is the equivalent hydraulic diameter of the channel (De 2b) and the parameters C3, C4fi 0:008 1 C3 hccDe

C4" #A:16obtain an independent estimate of interfacial friction factor, fi, the following modied empiricalcorrelationoriginally proposed by Bharathan et al. (1979)presented by Sudo (1996) for thecase of counter-current ow inside a vertical rectangular channel, may be employed:h is neglected. To the authors best knowledge, there are no data available on the value offacial shear stress in counter-current ow inside a vertical rectangular channel. In order toHere it is assumed that the supercial gas velocity is much greater than the liquid surface velocitysi 2fiqGU

2GS A:15The interfacial shear stress is expressed in terms of an interfacial friction factor, fi, as follows:ReGS UGSbqGlGA:14andfG 0:079Re0:25GS A:13In the above expressions, the gas wall shear stress, sG, is given in terms of a friction factor, fG, andthe supercial gas velocity, UGS; i.e.,

sG 12fGqGU

2GS A:12

wheresw;mean lLdUxdy

y0

qLg qGg 2

wsG 2arb si

hcc si A:111 qL qGg 2w sG 2arb si

qLghcchff

3

32

siqLghff

hcchff

2

A:10

The mean wall shear stress, sw,mean, is obtained from Eq. (A.6) for y = 0:ven that this analysis is restricted to counter-current ow below critical conditions, Q isdered constant independent of gas ow rate; dividing Eq. (A.8) by (A.9) leads toBoe

747754.Paras, S.V., Karabelas, A.J., 1991. Properties of the liquid layer in horizontal annular ow. Int. J. Multiphase Flow 17,

43Pierso

liqReiss,

80 E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181J. 8, 245247.9454.n, F.W., Whitaker, S., 1977. Some theoretical and experimental observations of the wave structure of fallinguid lms. Ind. Eng. Chem. Fund. 18, 401408.L.P., Hanratty, T.J., 1962. Measurement of instantaneous rates of mass transfer to a small sink on a wall. AIChEand Boe is the Bond number based on De:

Boe De qL qGgr 1=2

A:19

References

Alekseenko, S.V., Nakoryakov, V.E., Pokusaev, B.G., 1994. Wave Flow of Liquid Films. Beggel House Inc., NewYork.

Banko, S.G., Lee, S.C., 1986. A critical review of the ooding literature. In: Hewitt, G.F., Delhaye, J.M., Zuber, N.(Eds.), Multiphase Science and Technology. Hemisphere, New York, pp. 95180, Chapter 2.

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E.I.P. Drosos et al. / International Journal of Multiphase Flow 32 (2006) 5181 81

Counter-current gas - liquid flow in a vertical narrow channel mdash Liquid film characteristics and flooding phenomenaIntroductionExperimental setup and proceduresExperimental conditionsFilm thickness measurementsWall shear stress measurementsVisual observations

Visual observationsCritical flooding velocitiesInterpretation of instantaneous film thickness dataFilm thickness characteristicsPower spectral density of film thickness fluctuationsWave celerity

Interpretation of instantaneous wall shear stress dataMean and RMS wall shear stressPower spectral density of wall shear stress fluctuationsFilm thickness mdash wall shear stress interrelationship

Concluding remarksAcknowledgementA model to estimate mean film thickness and mean wall shear stress in counter-current gas ndash liquid flowReferences