Axial dispersion in segmented gas-liquid flow: Effects of alternatingchannel curvature

Metin Muradoglua�

Department of Mechanical Engineering, Koc University, Rumelifeneri Yolu Sariyer, Istanbul 34450, Turkey

�Received 31 March 2010; accepted 6 December 2010; published online 30 December 2010�

The effects of channel curvature on the axial dispersion in segmented gas-liquid flows are studiedcomputationally in a two-dimensional setting using a finite-volume/front-tracking method. Passivetracer particles are used to visualize and quantify the axial dispersion. The molecular diffusion ismodeled by random walk of tracer particles. It is found that there is significant axial dispersion inserpentine channels even in the absence of molecular diffusion. The lubricating thin liquid layer thatpersists on the wall of a straight channel is periodically broken in the serpentine channel leading toenhanced axial dispersion. It is also found that the axial dispersion is always larger in the serpentinechannel than that in the straight channel but the effects of channel curvature are more pronouncedat high Peclet numbers, i.e., Pe�104. A model is proposed based on the difference between theliquid film thicknesses on the inner and outer side of the bend in the limit as Pe→�. Goodagreement is found between the computational results and the model when the liquid slug is wellmixed by the chaotic advection. © 2010 American Institute of Physics. �doi:10.1063/1.3531742�

I. INTRODUCTION

Segmented gas-liquid flow �also known as Taylor flow�has been studied extensively since the pioneering worksof Taylor1 and Bretherton2 due to its significance in a numberof engineering processes such as two-phase catalyticmonolith reactors,3–6 polymer blow molding, enhanced oilrecovery,7 continuous-flow analysis of biological or chemicalsamples,8–10 and rapid mixing of solutes in microchannels.11

The micromixer developed by Günther et al.11 is based onthe chaotic stirring within the liquid slugs moving through agas-segmented serpentine channel and it has been shown toreduce the mixing lengths two- to threefold as compared topassive mixers with patterned side walls or three dimen-sional channel geometries. In addition to the enhanced mix-ing, the segmentation also reduces the axial dispersion sig-nificantly compared to the single phase flow and onlydispersion occurs due to either convection through liquidfilm between the channel wall and bubble or diffusionthrough the gas-liquid interface. Note that the liquid regionbetween two centroids of bubbles is referred as slug whilethe recirculating region in the liquid slug is referred as bolusin the present study.

The quality of mixing within the liquid slugs stronglydepends on the segmentation and the channel curvature asrecently discussed by Dogan et al.12 In the case of a straightchannel, there are two contour rotating symmetrical steadyvortices within the liquid slug in the coordinate system mov-ing with the bubbles. Thus there is no cross mixing in thestraight channel in the absence of molecular diffusion.12,13

This symmetry is broken in the case of a serpentine channeland mixing is enhanced by the channel curvature.12,13 How-ever, the channel curvature also increases the axial dispersion

that is usually unwanted in many applications. It is knownthat a lubricating thin liquid layer persists on the channelwalls when the channel is straight. However this lubricatingliquid layer is periodically broken in the serpentine channelcase leading to enhanced axial dispersion. Muradoglu andStone14 showed that the film thickness on the inner wall isthinner than that on the outer wall and this difference in filmthicknesses increases with channel curvature. As the liquidslugs move through a serpentine channel, the film thicknessalternates periodically due to change in the channel curvatureand thus significant amount of liquid leaks through the outerlayer even in the absence of the molecular diffusion. Al-though axial dispersion is still much reduced in the seg-mented gas-liquid flow compared to the corresponding singlephase flow, there may be significant backmixing especially inthe case of large channel curvatures. This backmixing en-larges the residence time distribution and thus deterioratesperformance of microreactors.15–18 It is therefore of funda-mental importance to understand and control the axial dis-persion in gas-segmented serpentine channels.

The early studies on the gas-segmented flow weremainly motivated by the continuous-flow analyzers.8 Re-cently, the gas-segmented flows have been mainly studieddue to applications in monolith reactors6,19–21 and enhancedmixing in microchannels.11,15 Pedersen and Horvath22 pro-posed a simple model for the axial mass transfer in gas-segmented straight cylindrical channel based on the assump-tions that there is a perfect mixing in each liquid slug andmass transfer from the recirculating bulk liquid region to thefilm region can be characterized by an adjustable mass trans-fer coefficient. Bercic and Pintar3 developed a computationalmodel for the effects of gas bubbles and liquid slug length onthe backmixing in the gas-segmented flow and found that themass transfer mainly depends on the slug length and velocity.Thulasidas et al.21 improved two-region model of Pedersen

a�Electronic mail: mmuradoglu@ku.edu.tr. Telephone: �90 �212� 338 14 73.Fax: �90 �212� 338 15 48.

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and Horvath22 by solving one-dimensional diffusion equationin the radial direction to account for the mass transfer bymolecular diffusion. Salman et al.23,24 developed a similarmodel based on the one-dimensional convection-diffusionequation. More recently they developed a computationalfluid dynamics model for the axial dispersion and residencetime distribution in a straight two-dimensional gas-segmented channel flow.25 Muradoglu et al.26 studied theaxial dispersion in the gas-segmented flow using a finite-volume/front-tracking �FV/FT� method and examined the ef-fects of the Peclet number, capillary number, and segmentsize in a straight two-dimensional channel. Kreutzer et al.27

studied the axial dispersion in a straight rectangular micro-channel and found that the liquid-filled menisci significantlyincrease the leakage and the axial dispersion is the leastwhen the liquid flow rate is greater than the gas flow rate.Based on the work of Muradoglu and Stone,14 Fries and vonRohr13 and Dogan et al.12 predicted that the channel curva-ture enhances the axial dispersion due to alternating filmthickness in serpentine channels but they did not provide anydetailed analysis.

In this paper, the effects of channel curvature on theaxial dispersion are studied computationally using the finite-volume/front-tracking method in a two-dimensional settingto facilitate extensive numerical simulations. Note that mi-crochannels usually have square or rectangular cross-section.Significant portion of leakage occurs through the corners ofthe rectangular channels at low Peclet numbers.27 Howeverthe effects of large gutter in rectangular channel are expectedto be small at high Peclet numbers. In addition, three dimen-sional microchannels of rectangular cross-section becomequasi-two-dimensional when the aspect ratio of the channelcross-section is large, i.e., it is larger than eight.28 Thereforethe main conclusions of the present study are expected to bevalid for the three dimensional rectangular channels at highPeclet numbers especially when the aspect ratio of the chan-nel cross-section is large. Passive tracer particles are used tovisualize and quantify the mass transfer. Molecular diffusionis modeled by a random walk of tracer particles in a similarfashion as done by Muradoglu et al.26 The effects of themolecular diffusion are characterized by the Peclet numberdefined as Pe=Vcw /D, where Vc, w, and D are the averageliquid velocity in the channel, the channel width, and themolecular diffusivity, respectively. Extensive simulations arefirst performed to show and quantify the effects of the chan-nel curvature on the axial dispersion in the absence of mo-lecular mixing, i.e., as Pe→�. Based on the alternating filmthickness in the serpentine channel,14 a simple analyticalmodel is developed and compared with the numerical results.Reasonable agreement is found between the analytical modeland the computational simulations. The computations arethen performed to show the effects of the molecular diffu-sion. It is found that the axial dispersion is always larger in aserpentine channel than that in the corresponding straightchannel but the effects of channel curvature are more pro-nounced at high Peclet numbers, i.e., Pe�104. It is alsofound that the difference between the straight and curvedchannels decreases continuously as the Peclet number de-

creases and becomes negligible at low Peclet numbers, i.e.,Pe�103.

The rest of the paper is organized as follows. The math-ematical formulation and numerical method are briefly sum-marized in the next section. The physical problem is de-scribed in Sec. III. The results are presented and discussed inSec. IV where the model is also described. Finally conclu-sions are drawn in Sec. V.

II. FORMULATION AND NUMERICAL METHOD

The governing equations are described here in the frame-work of the finite-volume/front-tracking method.29–31 In theCartesian coordinates, the two-dimensional incompressiblecontinuity and Navier–Stokes equations can be written inconservation form as

�q

�t+

�f

�x+

�g

�y=

�fv

�x+

�gv

�y+ fb, �1�

where

q = � 0

�u

�v�, f = � u

�u2 + p

�uv�, g = � v

�uv

�v2 + p� , �2�

and

fv = � 0

�xx

�xy�, gv = � 0

�xy

�yy� . �3�

In Eqs. �1�–�3�, x and y are the Cartesian coordinates and t isthe time; �, �, and p are the fluid density, the dynamic vis-cosity, and pressure, respectively, and u and v are the veloc-ity components in x and y coordinate directions, respectively.The first row in Eq. �1� simply states that the velocity field issolenoidal while the last two rows represent the momentumconservation equations in the x and y directions, respectively.The viscous stresses appearing in the viscous flux vectors aregiven by

�xx = 2��u

�x, �yy = 2�

�v�y

, �xy = �� �u

�y+

�v�x . �4�

The last term in Eq. �1� represents the body forces resultingfrom surface tension and is given by

fb = S

n��x − x f�ds , �5�

where �, x f, , , n, S, and ds denote, respectively, the Diracdelta function, the location of the interface, the surface ten-sion coefficient, twice the mean curvature, the outward unitnormal vector on the interface, the surface area of the inter-face, and the surface area element of the interface.

In Eq. �1�, it is assumed that the material properties of afluid particle remain constant, i.e.,

D�

Dt= 0;

D�

Dt= 0, �6�

where the substantial derivative is defined as

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D

Dt=

�

�t+ u · � .

The governing equations �Eq. �1�� are solved by the finite-volume/front-tracking method developed by Muradoglu andKayaalp.31 The method combines a finite-volume solver withthe front-tracking method developed by Unverdi andTryggvason.29 The continuity and momentum equations aresolved on a curvilinear grid using a finite-volume method.The spatial derivatives are approximated by a finite-volumemethod that is equivalent to second-order finite differenceson a regular mesh. A dual �or pseudo� time-stepping methodis employed to achieve time accuracy and an alternating di-rection implicit �ADI� method is used to perform integrationin pseudotime. Fourth-order numerical dissipation terms areadded to the discrete version of the flow equations to preventthe odd-even decoupling. Preconditioning, local time-stepping, and multigrid methods are used to accelerate theconvergence rate of the ADI method in the pseudotime. De-tails of the FV method can be found in Refs. 31 and 32.

The interface boundary between the bubble phase andthe ambient fluid are represented by connected Lagrangianmarker points moving with the local flow velocity interpo-lated from the neighboring curvilinear grid points. The com-munication between the curvilinear grid and the interfacemarker points is maintained efficiently using an auxiliaryregular Cartesian grid cast on the curvilinear grid.31 An indi-cator function is defined such that it is unity inside the drop-let and zero outside. The indicator function is computed ateach time step based on the locations of the interface markerpoints as described by Tryggvason et al.30 and Muradogluand Kayaalp.31 Once the indicator function distribution isdetermined, the density and viscosity are set as a function ofthe indicator function. The interface marker points arealso used to compute the surface tension forces at the inter-face which are then distributed on the neighboring curvilin-ear grid cells in a conservative manner and added to thediscrete momentum equations as source terms. For the de-tails of the FV/FT method, see Muradoglu and Kayaalp31 andMuradoglu and Gokaltun.33

Passive tracer particles are used to visualize and quantifythe axial dispersion in a similar way as done by Muradogluet al.26 The molecular mixing is modeled by random walk oftracer particles. In a Lagrangian frame, the location of mthparticle Xm evolves by

dXm = Umdt + �2DdW , �7�

where Um is the velocity interpolated from the Eulerian cur-vilinear grid onto the location of the tracer particle, dt is thetime increment, D is molecular diffusivity of the tracer, anddW is a vector valued Wiener process34 with the well-knownproperties of �dW =0 and �dWidWj =�ijdt, where �¯ de-notes the mean and �ij is the Kronecker delta. Equation �7� issolved numerically using a predictor-corrector method to-gether with the flow equations as described by Muradogluet al.26 The tracer particles crossing the channel wall or thebubble interfaces are reflected back into liquid region assum-ing a perfect collision. Note that the indicator function isutilized to reflect the particles crossing the bubble interfaceas described by Muradoglu et al.26 while reflection isachieved using the geometric information for the particlescrossing the channel walls.

III. PROBLEM STATEMENT

The effects of the channel curvature on the axial disper-sion in gas-segmented flow in a serpentine channel are stud-ied in a two-dimensional setting to facilitate extensive com-putational simulations. The model channel consists of astraight entrance, a curved mixer of three periods, and astraight exit section as sketched in Fig. 1. The curved sectionof the mixer consists of concentric circular arcs and the innercircle of a half period connects tangentially with the outercircle of the next half period. The diameters of the inner andouter arcs are Di and Do, respectively. The channel width isthen given by w= �Do−Di� /2. Finally we complete the speci-fication of the geometry by fixing Li, the length of the inletportion of the channel, Le, the length of the exit portion ofthe channel, and Lm, the length of the mixing portion of thechannel as sketched in Fig. 1. The inlet portion of the chan-nel that contains the gas bubbles and the initial distributionof the tracer particles are shown in Fig. 2. The flow rate isspecified at the inlet assuming a fully developed velocityprofile with an average velocity of Vc. The flow is initializedas follows: a steady single-phase flow is computed first usingthe liquid properties and is then used as initial flow field. Thebubbles are instantaneously placed in the channel close to theinlet with an approximate shape consisting of a straightmiddle and semicircular front and back sections. Passive

L

Lm

Li

Le

Di

Do

FIG. 1. The sketch of the channel used in the computations. The inset showsa portion of coarse grid containing 600�32 grid cells.

GasGas

Lo

Tracer Particles

h

w

Liquid

Vc

Ls

FIG. 2. A schematic illustration of the inlet portion of a two-dimensionalchannel containing two bubbles and one liquid segment. A large number oftracer particles is distributed at random in the liquid segment after gasbubbles reach their steady shapes.

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tracer particles are initially distributed uniformly at randomin the channel outside of the bubbles when the bubbles taketheir steady shapes in the inlet section of the channel asshown in Fig. 2. The particles are moved with the local flowvelocity interpolated from the neighboring computationalgrid points using the same advection scheme as used formoving the interface marker points.

The properties of gas bubble and liquid are denoted bysubscripts i and o, respectively. The governing nondimen-sional parameters are defined as the channel Reynolds num-ber Re=�oVcw /�o, the capillary number Ca=�oVc /, thePeclet number Pe=Vcw /D, the viscosity ratio =�i /�o, thedensity ratio r=�i /�o, the normalized distance between thegas bubbles in the inlet section �=Lo /w, the normalized cur-vature of the inner channel wall �=2w /Di, and the normal-ized equivalent bubble diameter �=db /w where db is theequivalent bubble diameter. The nondimensional time is de-fied as t�= tVc /w. Note that all the computations presented inthis paper are performed at Re=0.64, =0.1, r=0.1, �=1.5,and �=0.667. It is emphasized here that these viscosity anddensity ratios are much larger than the typical values encoun-tered in microfluidic applications. For instance, �0.01 andr�0.001 for ethanol/air or water/air systems. However, theeffects of viscosity and density ratios are expected to besmall when �0.1 and r�0.1 for the kind of problems stud-ied here.14,26

The averaged tracer concentration in the liquid slug isdefined as

�C =N

As, �8�

where N is the total number of particles in the liquid slugbetween the centroids of the bubbles and As is the area of theslug. The slug area is assumed to remain constant throughoutthe simulation so that the normalized average tracer concen-tration is simply computed as

�C �C i

=N

Ni,

where Ni is the initial number of tracer particles in the slug.

IV. RESULTS AND DISCUSSION

Extensive computational simulations are performed toexamine the effects of the channel curvature on the axialdispersion. A portion of a typical �coarse� curvilinear gridcontaining 600�32 grid cells is shown in the inset of Fig. 1to show the overall grid structure. The computations are per-formed on a finer version of this grid that contains 2400�128 grid cells. As can be seen in Fig. 1, the grid isstretched near the channel walls in order to resolve the thinliquid films between the bubbles and the channel. The gridconvergence of the present numerical method has been ex-tensively examined and this grid resolution has been shownto be sufficient to achieve grid independent solutions for thiskind of problems as long as Ca�0.005.12,14,26,35 Thereforesuch an extensive grid convergence study is not repeatedhere. Note that the capillary number is often smaller than0.005 in microfluidic applications11,27 but the simulations

are limited to this value in the present study mainly due tothe excessive computational cost required to resolve thin liq-uid film region that is of critical importance for the axialdispersion.

First, the molecular diffusion is switched off in order toexamine the sole effects of the channel curvature on the axialdispersion. For this purpose, the scatter plots of the tracerparticles are plotted in Fig. 3 for the straight and curvedchannel cases. As can be seen in this figure, a lubricating thin

(a)

(b)

(c)

FIG. 3. �Color online� Scatter plots of the tracer particles showing the en-hancement of the axial dispersion due to the channel curvature in the ab-sence of the molecular diffusion. Different colors are used for the particlesinitially located in the lower and upper portions of the liquid segment toshow the mixing patterns as well as the axial dispersion. There is no axialdispersion in the case of the straight channel �a� but there is significant axialdispersion for the case of the curved channel �b� in the convex and �c�the concave segments �Ca=0.01, Re=0.64, =0.1, r=0.1, �=1.5, and�=0.667�.

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liquid film forms on the wall of the straight channel andpersists as the bubbles pass leading to no axial dispersion. Inthe case of serpentine channel, however, this lubricating liq-uid film is broken periodically resulting in significant axialdispersion as seen in Figs. 3�b� and 3�c�. It is also seen in thisfigure that there is still a thin lubricating liquid film even inthe case of the serpentine channel and leakage occurs due toalternating film thickness. It is seen visually from these scat-ter plots that the liquid layer with an approximate width of�h=ho−hi leaks in every half period where hi and ho are thefilm thickness on the inner and outer channel walls, respec-tively. The leakage occurs through the liquid film on theouter wall, once on the one side and then on the other side ofthe channel as shown in Fig. 3. As recently discussed byDogan et al.,12 in the coordinate system moving with thecentroid of the liquid slug, the velocities at the inner andouter wall can be approximated by Ut=2�1+��Vb / �2+��and Ub=2Vb / �2+��, respectively, and the flow in the liquidslug resembles a double-lid-driven cavity flow with liquidleakage through the edges at small capillary numbers. Usingthis analogy, each half period of the curved channel can bemapped into a straight channel segment with alternating liq-uid film thickness and wall velocity as sketched in Fig. 4. Inthe first half period, the leakage occurs through the liquidfilm on the upper wall and then through the liquid film on thelower wall in a periodic fashion. The amount of tracer thatleaks in each half period �hperiod can be approximated as�hLs�C where Ls is the length of the liquid layer �Fig. 2�.Assuming that the liquid slug is well mixed due to the cha-otic advection in each half period, the average concentrationin the liquid slug decays according to

�C �C i

= e−�t�, �9�

where �C i is the initial average concentration and

� =�Ah

As

1

�h period

w

Vc. �10�

In Eq. �10�, As is the area of the liquid slug, �Ah is the areaof liquid layer that leaks through the outer film in thehalf period. The area of the liquid layer is computed as�Ah=�hLs and the half time period is approximated as�h period=��Ri+w /2� /Vb, where Vb is the average bubble ve-locity. Based on the analysis due to Muradoglu and Stone,14

the film thicknesses on the inner and outer walls of thecurved channel can be computed as

hi = 1.3375ReffiCaeffi

2/3, ho = 1.3375ReffoCaeffo

2/3 , �11�

where

Reffi=

1

2 + �; Reffo

=1 + �

2 + �;

�12�

Caeffi=

2 + 2�

2 + �Ca; Caeffo

=2

2 + �Ca.

In Eq. �12�, �=2w /Di is the nondimensional channel curva-ture and Ca=�oVc / is the capillary number. Equation �11�is valid only for small capillary numbers, i.e., Ca�1. Forlarger capillary numbers, this equation can be modified as

hi,o

Reffi,o

=1.34 Caeffi,o

2/3

1 + 1.34 � 2.5 Caeffi,o

, �13�

which is based on the correlation suggested by Aussilous andQuere36 and has been also shown to agree very well with thenumerical simulations for a wide range of capillary numbersin curved channels.14 Note that microchannels usually havesquare or rectangular cross-section and a significant portionof leakage occurs through the corners of the rectangularchannels at low Peclet numbers.27 However the effects of alarge gutter in rectangular channel are expected to be small athigh Peclet numbers since the leakage is solely caused by thealterations in the liquid film thickness in the limit of vanish-ing molecular diffusion. In addition, three dimensional mi-crochannels of rectangular cross-section become quasi-two-dimensional when the aspect ratio of the channel cross-section is large, i.e., it is larger than eight.28 Therefore themain conclusions of the present study are expected to bevalid for the three dimensional rectangular channels at highPeclet numbers especially when the aspect ratio of the chan-nel cross-section is large.

The theory developed above is tested against the numeri-cal simulations. For this purpose, the computations are per-formed for two different capillary numbers, i.e., Ca=0.01and Ca=0.005. The relative distance between bubbles isabout Lo /w=0.62 and Lo /w=0.64 for Ca=0.01 and Ca=0.005, respectively. The evolution of average tracer con-centration is plotted in Fig. 5 where Eq. �9� is shown asdashed curves. Since the theory is valid only after the initial

top wall

bottom wall

Ut

Ub

bubble

ho

hi

�h = ho - hi

bubble

Liquid layerto leak

(a)

top wallUt

bottom wallUb

bubble

ho

hi

bubble

�h = ho - hi

Liquid layerto leak(b)

FIG. 4. �Color online� Schematic illustration of the model for the leakagethrough liquid films �a� at the top and �b� at the bottom of the channel.

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period in which the concentration is reduced just because thetracer particles originally in the film region are convecteddownstream �relative to the coordinate system fixed to abubble�, the time is started after this initial period in all thetheoretical results presented here. As can be seen in this fig-ure, there is significant axial dispersion in curved channelsand the leakage increases with increasing capillary number.On the other hand, there is no axial dispersion in the straightchannel case except for the initial period as also shown byMuradoglu et al.26 The theoretical results are in a reasonablygood agreement with the computational simulations for bothcapillary numbers. The discrepancy between the theoreticaland computational results can be partly attributed to the un-certainties in computing slug area and slug length used inEq. �10�. It is interesting to observe that the decay rate isoverpredicted for Ca=0.01 while it is underpredicted forCa=0.005, and the theoretical prediction is in a better agree-ment with the computational result for Ca=0.005. Note thatthe bumpy feature observed in the evolution of the averageconcentration in serpentine channel is mainly caused by therapid change in liquid film thickness as a result of change inthe channel curvature when the bubbles move from one halfperiod to the next. Consequently, a large amount of tracerparticles is convected rapidly through the enlarged liquidfilm at the beginning of each half period.

Next, the effects of the relative distance between bubblesare investigated. For this purpose, the molecular diffusion isagain switched off and the computations are performedfor three values of relative distance between bubbles atCa=0.01 and Ca=0.005. Note that the uncertainties in com-puting slug area and slug length decrease as the distancebetween the bubbles increases. But, at the same time, thequality of mixing deteriorates as the slug length increases asdiscussed recently by Dogan et al.12 The computational re-sults are plotted in Figs. 6�a� and 6�b� for Lo /w=0.62, 0.79,

and 1.37 at Ca=0.01 and for Lo /w=0.64, 0.89, and 1.4 atCa=0.005, respectively. The computational results are com-pared with the theory for all the cases. The concentrationdecay rate is consistently overpredicted for Ca=0.01 while itis underpredicted for Ca=0.005 for all values of Lo /w. Fig-ure 6 also shows that the decay rate decreases with increas-ing Lo /w just because the slug size gets larger and thus ittakes a longer time for the tracer particles to leak through theliquid film.

Finally the effects of the molecular diffusion are inves-tigated. For this purpose, the Peclet number is varied in therange between Pe=100 and Pe→� and computations arerepeated for Ca=0.01 and Ca=0.005 in both the straight andserpentine channel cases. The molecular diffusion is ex-pected to increase the axial dispersion due to the enhancedmass transfer from the slug to the wall region as well asimproved mixing within the liquid slug. The effects ofthe molecular diffusion on the axial dispersion have beenrecently studied extensively in straight channel case byMuradoglu et al.26 Therefore the emphasis is placed here onthe effects of the channel curvature in the presence of mo-

0 5 10 15 20 25 30 350.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Non−dimensional time, t*

⟨C⟩/⟨

C⟩ i

Straight Channel

Curved Channel

Ca = 0.005

Ca = 0.01

Theory

FIG. 5. �Color online� Effects of the channel curvature on the axialdispersion. The relative distance between bubbles are Lo /w=0.62 andLo /w=0.64 for Ca=0.01 and Ca=0.005, respectively. The computationalresults for the serpentine channel �solid lines� are compared with the theo-retical results �dashed lines�. The corresponding results for the straight chan-nels are also shown by dotted lines �Re=0.64, =0.1, r=0.1, �=1.5, and�=0.667�.

0 5 10 15 20 25 30 350.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Non−dimensional time, t*

⟨C⟩/⟨

C⟩ i

Lo/w = 0.62

Lo/w = 0.79

Lo/w = 1.37

Numerical Results

Theory

(a)

0 5 10 15 20 25 30 350.7

0.75

0.8

0.85

0.9

0.95

1

Non−dimensional time, t*

⟨C⟩/⟨

C⟩ i

Lo/w = 1.4

Lo/w = 0.89

Lo/w = 0.64

Numerical Results

Theory

(b)

FIG. 6. �Color online� Effects of the relative distance between bubbles onthe axial dispersion for �a� Ca=0.01 and �b� Ca=0.005. The theoreticalpredictions are shown by dashes lines �Re=0.64, =0.1, r=0.1, �=1.5, and�=0.667�.

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lecular diffusion. The time evolution of the tracer concentra-tion is plotted in Figs. 7�a� and 7�b� for Ca=0.01 andCa=0.005, respectively. As can be seen in these figures, theaxial dispersion increases monotonically with decreasingPeclet number both for the straight and curved channel cases.The axial dispersion is always larger in the serpentine chan-nel than that in the straight channel. However, the effects ofchannel curvature reduce as Peclet number decreases andbecomes negligible at very low Peclet numbers since theaxial dispersion is mainly controlled by the convectionthrough the liquid film in the low Peclet number limit asPe→0. On the other hand, the effects of channel curvatureincrease rapidly at high Peclet numbers, i.e., Pe�104, andthe axial dispersion is solely caused by alternating channelcurvature as Pe→�.

V. CONCLUSIONS

Effects of channel curvature on the axial dispersion ingas-segmented liquid flow are studied computationally usinga finite-volume/front-tracking method. A large number ofpassive tracer particles are used to visualize and quantify the

axial dispersion. Molecular diffusion is modeled by the ran-dom walk of the tracer particles. The particles crossing theliquid-solid and liquid-gas interfaces are reflected back intothe liquid region using a perfect elastic collision model. Geo-metric information is used to reflect the particles crossing thesolid wall while the indicator function is utilized for the par-ticles crossing the liquid-gas interfaces as described by Mu-radoglu et al.26 Computations are performed in a two-dimensional setting to facilitate extensive numericalsimulations.

The molecular diffusion is first switched off in order todemonstrate the axial dispersion solely due to the alternatingfilm thickness in a serpentine channel. It is found that thelubricating thin liquid layer that persists on the wall of astraight channel is periodically broken in the curved channelcase leading to significant axial dispersion. It is confirmedthat there is no axial dispersion in a straight channel in theabsence of molecular diffusion after the initial period inwhich the particles originally in the liquid film region aresimply convected downstream. A model is proposed for theenhancement of axial dispersion in serpentine channel basedon the difference between the liquid film thickness on theinner and outer channel walls. The model assumes a perfectmixing within the liquid slug in each half period, a conditionthat is usually well satisfied in applications due to chaoticstirring. The model predicts that concentration of the tracerwithin the liquid slug decays exponentially in time and thedecay rate is proportional to the difference between the liq-uid film thickness in successive half periods. It is found thatthe model predicts the decay rate of the tracer concentrationreasonably well compared to the computational results. Theaxial dispersion strongly depends on the capillary numberand increases as capillary number increases. The theoreticalprediction is found to be in a better agreement with the com-putational results for smaller capillary numbers. The effectsof the liquid slug length or equivalently the distance betweenthe gas bubbles are also investigated. It is found that thetracer concentration decays faster as the slug length de-creases. This is mainly due to the fact that it takes longertime for tracer within a large liquid slug to leak through thefilm and also partly due to the enhanced mixing in smallliquid slugs.

Then the combined effects of channel curvature and mo-lecular diffusion are studied. Computations are performedboth for the straight and serpentine channel cases for a rangeof Peclet numbers between Pe=100 and Pe→�. It is foundthat the effects of the channel curvature are more pronouncedat high Peclet numbers, i.e., Pe�104. The axial dispersion isprimarily controlled by the convection through the liquidfilms and thus the effects of the channel curvature are mar-ginal at low Peclet numbers, i.e., P�100. On the other hand,the axial dispersion is mainly controlled by the quality ofmixing due to chaotic advection and alternating film thick-ness in the high Peclet number limit, i.e., Pe�106. In be-tween, there is a transition regime where both the channelcurvature and molecular mixing are important.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Non−dimensional time, t*

⟨C⟩/⟨

C⟩ i

Pe increasing

Straight Channel

Curved Channel

(a)

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Non−dimensional time, t*

⟨C⟩/⟨

C⟩ i

Pe increasing

Straight Channel

Curved Channel

(b)

FIG. 7. �Color online� Effects of molecular diffusion. �a� Ca=0.01and Lo /w=0.62. �b� Ca=0.005 and Lo /w=0.64 �Re=0.64, =0.1, r=0.1,�=1.5, and �=0.667�.

122106-7 Axial dispersion in segmented gas-liquid flow Phys. Fluids 22, 122106 �2010�

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ACKNOWLEDGMENTS

The author thanks Professor Howard A. Stone for help-ful conversations. This work has been supported by theScientific and Technical Research Council of Turkey�TUBITAK� with Grant No. 108M238 and Turkish Academyof Sciences �TUBA-GEBIP�.

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