International Journal of Heat and Fluid Flow 33 (2012) 147–155

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International Journal of Heat and Fluid Flow

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A model of a bubble train flow accompanied with mass transfer through a longmicrochannel

Dmitry Eskin ⇑, Farshid MostowfiSchlumberger DBR Technology Center, 9450-17 Avenue, Edmonton, AB, Canada T6N 1M9

a r t i c l e i n f o

Article history:Received 6 April 2011Received in revised form 27 October 2011Accepted 1 November 2011Available online 25 November 2011

Keywords:BubbleLiquid slugMass transferMicro-channelMathematical modelingMixing

0142-727X/$ - see front matter � 2011 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2011.11.001

⇑ Corresponding author. Tel.: +1 780 577 1319; faxE-mail address: deskin@slb.com (D. Eskin).

a b s t r a c t

A model of a bubble train flow accompanied with mass transfer in a long capillary tube is developed. Incontrast to models presented in literature, our modeling approach accounts for expansion of gas bubblesand flow velocity increase along the channel due to the pressure drop caused by friction losses. The modelperformance is illustrated by a number of computational examples. The distributions of the bubble veloc-ity and the volumetric mass transfer coefficient along the channels of different diameters are presented.The deviation of the dissolved gas concentration from the saturation concentration along the channel isused as a characteristic of closeness of a fluid system to the phase equilibrium. The calculations explicitlydemonstrate that the deviations of gas–liquid mixture flows from equilibrium in long capillary channelsof small diameters are small. The effect of the channel diameter, the channel length, and the bubblenucleation frequency on the deviation of the system from equilibrium is also studied.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Capillaries and microchannels have been used as efficient masstransfer devices for some time. For example, monolithic catalystreactors demonstrate superior performance as mass exchangers(e.g. Irandoust and Andersson, 1989). In the present work we con-sider mass transfer only for a bubble train (Taylor) flow regimewhere cylindrical bubbles, separated by liquid slugs, move alongthe channel.

Note that the bubble train flow dynamics has been intenselystudied by a significant number of researches (e.g. Irandoust andAndersson, 1989; Laborie et al., 1999; Angeli and Gavrilidis, 2008).

The current work is focused on developing a model of a flowaccompanied with mass transfer in a long micro-channel to evalu-ate its performance as a mass exchanger in dependence on flowrate, channel diameter and length. We would like to emphasizethat our primary interest is in flows in micro-channels of diametersless than 200 lm and lengths of the order of a meter. The moderntechnologies allow for manufacturing such channels, which can beused as extremely efficient mass exchangers. We consider onlychannels with circular cross-sections. Although microchannels thathave rectangular cross sections are widely used in practical appli-cations, it is easier to model a flow in a circular capillary tube. It isalso possible to assume that the mass transfer process in a rectan-gular channel is similar to that in a circular capillary tube and that

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: +1 780 450 1668.

the corresponding mass transfer rates are not significantly differ-ent. Note also that we limit our work to two-component mixturesonly. The modeling approach to be presented can be extended tomulti-component mixtures, but this is beyond the scope of the cur-rent research.

The void (gas) fraction and the corresponding flow velocitychange significantly due to a large pressure drop along a long chan-nel. The mass-transfer rate between a gas and a liquid varies con-siderably along the channel due to an increase in the void fractionand the corresponding flow acceleration towards the channel out-let. For characterizing mass transfer, besides the volumetric masstransfer coefficient (see Eq. (32)) we will use the deviation of thedissolved gas concentration from its saturation concentration(see Eq. (33)). This criterion will be introduced to evaluate thedeviation of the gas–liquid mixture from phase equilibrium, whichis observed when the concentration of a dissolved gas equals itssaturation concentration. To nucleate bubbles in a flow, the pres-sure should be reduced well below the bubble point pressure(the saturation pressure) of a given mixture (see e.g., Lubetkin,2003). In other words, the bubble train flow may initially be understrongly non-equilibrium conditions. The gas–liquid mixture mov-ing along the channel rapidly approaches equilibrium due to theefficient gas–liquid mass transfer.

Thus, bubbles can only be generated in a supersaturated liquidwhere the dissolved gas concentration is noticeably higher thanthe saturation concentration corresponding to the saturation pres-sure. As bubbles move along the channel, they grow for the follow-ing reasons: (1) release of the dissolved gas from the liquid due to

Nomenclature

Ca capillary numberc mass concentration of a dissolved gas, kg/m3

cm mass concentration of a dissolved gas, averaged over thefilm thickness, kg/m3

c0 mass concentration of a dissolved gas in a slug, kg/m3

cs saturation gas concentration, kg/m3

D molecular diffusivity of a dissolved gas, m2/sDB bubble diameter, mDc capillary channel diameter, mf bubble generation frequency, HzH Henry constant, m2/s2

kLa volumetric mass transfer coefficient, 1/skL,cap mass transfer coefficient through a side cap, m/sLB bubble length, mLlB length of cylindrical bubble part, mLsl slug length, mM bubble mass, kgp pressure, PaQG gas flow rate, m3/sQL liquid flow rate, m3/sq mass flux from liquid to gas, kg/sqs mass flux through the cylindrical bubble part, kg/sR gas constant, J/(kg K)Re Reynolds numberRB bubble radius, mRc capillary channel radius, mT temperature, KUB bubble velocity , m/s

UGS gas superficial velocity, m/sULS liquid superficial velocity, m/sUm mean flow velocity, m/sWB bubble volume, m3

Ws volume of a liquid associated with a bubble–slug pair,m3

Wse liquid slug volume, m3

Greek symbolsD relative concentration deviationDp/L mean pressure gradient, Pa/md liquid film thickness, me gas volume fractioneL liquid volume fractionc interfacial tensionlf dynamic fluid viscosity, Pa sq density, kg/m3

Subscriptsbf back capcf frontal capG gasB bubblei bubble or slug numberL liquidsl slugR total

148 D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155

the reduction in the gas solubility; (2) gas expansion due to thepressure reduction. A bubble in a bubble train flow has a cylindri-cal shape and is constrained by nearly semi-spherical surfaces onthe sides. Mass transfer in such a flow is carried out by both con-vection and diffusion through the gas–liquid interfaces. Theseinterfaces can be considered to be comprised of the cylindrical sur-face and the semi-spherical caps at both ends of a bubble. Note thatthe bubble train flow is characterized by two counter-rotating vor-tices (see e.g., Irandoust and Andersson, 1989), which provide anefficient convective mass transfer through the side caps.

There are many publications where mass transfer in the bubbletrain capillary flow is investigated experimentally and numerically(e.g., Irandoust et al., 1992; Van Baten and Krishna, 2004; Bercicand Pintar, 1997).

For our model, we employed some elements known from liter-ature and from two important publications in particular, whichpresent relatively simple models of mass transfer: Irandoustet al. (1992) and Van Baten and Krishna (2004). In Irandoustet al. (1992) a model of mass transfer for a Taylor flow in a verticalcapillary is presented, in which the authors used an analytical solu-tion of the convection–diffusion equation for a cylindrical thin li-quid film and empirical correlations for calculating the masstransfer through the semi-spherical caps. Perfect mixing within aliquid slug was assumed. The latter assumption means that the dis-solved gas concentration is uniform over the slug volume except atthe gas–liquid interface, where the phase equilibrium occurs. Theauthors also experimentally investigated mass transfer (oxygenabsorption into three different liquids) in vertical capillary tubesof the two different diameters (Dc = 1.5 and 2.2 mm). Comparisonof the simulation results with the measured data showed thatthe computations significantly overestimated the volumetric masstransfer coefficient. The authors explained this deviation by thetwo possible causes: (1) the model assumes perfect mixing in a

liquid slug, while in practice, mixing is incomplete; (2) accumula-tion of impurities on the gas–liquid interface makes the interfacemore rigid that reduces the mass transfer rate. Van Baten andKrishna (2004) also investigated a mass transfer in flows throughvertical capillaries of different diameters (Dc = 1.5, 2, and 3 mm)and developed a model, similar to that of Irandoust et al. (1992).For calculation of the mass transfer through the liquid film theyused an analytical solution for the falling liquid film that is ex-pected to provide almost the same results as the convection–diffu-sion equation used by Irandoust et al. (1992). However, Van Batenand Krishna (2004) employed a different semi-empirical correla-tion for calculating mass transfer through the side caps. Perfectmixing in a liquid slug was also assumed. In contrast to the resultsof Irandoust et al. (1992), the comparison of the mass transfer coef-ficients calculated by the engineering model of Van Baten andKrishna with those computed by the commercial CFD code (CFX)demonstrated very good agreement.

We would like to also emphasize that a practical objective ofthe current work is development of a modeling tool for preliminaryinvestigation of performance of a microfluidic based mass exchan-ger. This tool has to be suitable for estimations of optimal dimen-sions and potential efficiency of this mass exchanger, by avoidingcomplicated and costly experiments. To the best of our knowledge,the only possible way to reliably evaluate the mass transfer in amicrochannel experimentally is to analyze chemical compositionsof fluid samples taken at different points along the channel. How-ever, sampling from a microchannel of an extremely small diame-ter (25–200 lm in this paper), operating under high pressure (tensof Bars), is a very difficult problem. We are not aware of any exper-iments of such a kind presented in the open literature. One mightthink that an image analysis of flow patterns in a microchannel canbe used for mass transfer analysis. However, based on the results,presented in a recent paper of Molla et al. (2011), it was concluded

D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155 149

that flow visualization cannot provide valuable information on themass transfer rate in a microchannel. The authors of that paperinvestigated bubble-train flows in a long (�0.8 m) microchannellof a rectangular cross-section (50 � 100 lm). The three gas–liquidsystems (water–nitrogen, dodecane–nitrogen, and pentadecane–nitrogen) were employed for the experiments. Calculations, per-formed by neglecting the mass transfer, showed that the predictedbubble lengths were in a good agreement with those obtained fromthe image analysis of flow patterns in the channel. Under condi-tions of the high pressure drop in microchannels, a contributionof mass transfer into a bubble size increase is small compared toa contribution of a gas expansion, caused by pressure reduction,and cannot be reliably estimated by the flow visualization. The dif-ficulties of conducting mass transfer experiments in long micro-channels cause a need in an accurate modeling tool for designand optimization of microchannel based mass exchangers andreactors.

2. The model of a bubble train flow accompanied with masstransfer

The model of a flow accompanied with mass transfer, developedwithin this work, consists of a number of components required todescribe the entire process. Because in contrast with the formerinvestigations the current research is focused on modeling a flowin a long microchannel, we need to accurately simulate an evolu-tion of major flow characteristics along such a channel: the pres-sure, the velocity of a bubble, the mean velocity of a liquid slug,and the bubble size.

Let us formulate the restrictions, which we impose on a two-phase flow, analyzed in the current work. We will consider a sys-tem characterized by small mass fraction of a gas dissolved in anincompressible liquid; therefore, the volume flow rate of a liquidis assumed to be constant along the channel. We also limit ourmodel to isothermal flows. This assumption is plausible because gi-ven that the mass of a fluid in a micro-channel is very small, itstemperature will be close to ambient temperature. Thus, we canconfidently assume that all the physical parameters of a liquidare constant along the channel.

We will begin the model description with a bubble size evolu-tion that is a complex function of the major flow parameters andthe gas–liquid mass transfer rate.

2.1. Bubble size evolution

The slugs and bubbles in a steady-state Taylor flow in a micro-channel can be sequentially numbered as: 1, . . . i � 1, i, i + 1, . . . (seeFig. 1). Note that we distinguish bubbles and slugs by numbers be-cause the ith bubble is in contact with both the ith and the i + 1thslugs, which are characterized by different dissolved gas concen-trations and different gas–liquid mass transfer rates respectively.

Fig. 1. Diagram of a b

Also, the ith slug is in contact with both the i � 1th and the ith bub-bles. Due to the pressure drop along the slug, the saturation con-centrations are different at its sides, causing different masstransfer rates to the contacting bubbles. Thus, for mass transfermodeling, frontal and back sides of a bubble as well as those of aslug should be distinguished.

Two separate phenomena contribute to the rapid expansion ofbubbles: pressure drop and mass transfer from liquid phase.Growth of the ith bubble can be described by a simple mass bal-ance equation:

dMi

dt¼ qRi ð1Þ

where Mi ¼ .GiWBi is the bubble mass, WBi is the bubble volume, .Gi

is the gaseous phase density, and qRi is the gas mass flux from theliquid to the bubble.

By taking into account the expression for the bubble mass Mi

from Eq. (1), we easily obtain the differential equation describingthe bubble volume evolution in time:

dWBi

dt¼ qRi �

d.Gi

dtWBi

� �1.Gi

ð2Þ

Because the flow is steady-state, this equation can be rewrittenin terms of the axial coordinate x as:

dWBi

dx¼ 1

UBiqRi � UBi

d.Gi

dxWBi

� �1.Gi

ð3Þ

where dx = UBidt and UBi is the bubble velocity.We will assume that the gaseous phase behavior obeys the ideal

gas law as this assumption often works for two-component sys-tems if the maximum pressure is significantly smaller than thecritical pressure. For example, the ideal gas law is reasonably validfor methane (CH4) – decane (C10H22) system if the pressure doesnot exceed 500 psi (�3.4 MPa). Therefore, the gaseous phase den-sity is:

.Gi ¼pi

RTð4Þ

where pi is the pressure inside the ith bubble, R is the gas constant,and T is the temperature.

After substitution of Eq. (4) into Eq. (3) we obtain:

dWBi

dx¼ RT

pi

qRi

UBi� 1

pi

DpL

� �i

WBi ð5Þ

where DpL

� �i is the mean pressure gradient, determined as the ratio of

the pressure drop along the ith ‘‘gas bubble + liquid slug’’ pair to thelength of this pair (see e.g. Kreutzer et al., 2005).

As was recently proven for a Taylor flow (see Appendix A), themean pressure gradient is nearly constant along a microchannel.This means that the pressure linearly decreases in a stream-wisedirection as:

δ

ubble train flow.

Fig. 2. Diagram of a liquid film flow in coordinate system moving with the bubblevelocity.

150 D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155

p ¼ p0 �p0 � pe

Lx ð6Þ

where L is the microchannel length and p0 and pe are the inlet andoutlet pressures respectively.

Note that for the channel sizes we consider in this work, thebubble capillary pressure is negligibly small compared to the abso-lute pressure. Then the pressure pi inside the bubble can be calcu-lated by Eq. (6). The mean pressure gradient Dp

L

� �i is constant and

equal to � p0�peL . It should be noted that the pressure drop along a

bubble is small when compared to the pressure drop along a liquidslug (e.g., according to the CFD simulations of Kreutzer et al., 2005).

As one can see, the bubble velocity UBi is one of the majorparameters determining dynamics of bubble growth along thechannel. Let us calculate this velocity. Initially, we will formulatethe relation between the velocity of the bubble and the meanvelocity of the slug adjacent to this bubble. It is well known thatthe liquid film separating the cylindrical section of the bubbleand the wall is stationary (e.g., Bretherton, 1961; Aussillous andQuere, 2000). Then from the equality of the volume flow ratesthrough the neighboring channel cross-sections, occupied by the li-quid slug and by the bubble respectively, we obtain the equationfor the bubble velocity as:

UBi ¼Usli

1� diRc

� �2 ð7Þ

where Usli is the mean velocity of the ith slug, and di is the thicknessof the liquid film separating the ith bubble and the channel wall.

It is clear that the mean slug velocity equals to the mean flowvelocity Umi, which is constant along the ith ‘‘bubble + slug’’ pair.Then the slug velocity for a given channel cross-section is calcu-lated, based on the superficial velocity of a liquid and the void frac-tion at that cross-section:

Usli ¼ULS

1� eið8Þ

where ULS ¼ QLF is the superficial velocity of a liquid, F is the area of

the channel cross-section, QL is the liquid volume flow rate, and ei isthe void fraction for the ith ‘‘bubble + slug’’ pair.

The void fraction is calculated as:

ei ¼WBi

Ws þWBið9Þ

where Ws is the total (constant) liquid volume localized within a‘‘bubble + slug’’ pair.

We would like to note that we calculate the void fraction di-rectly, from the mass balance for a slug flow (Eq. (9)). There aresome publications (e.g. Kawahara et al., 2002), where the authorsstudying flows in microchannels of very small diameters (100 lmand smaller) demonstrated differences between calculated(homogenous) and measured (time averaged) void fractions.Kawahara et al. (2002) reported that the time averaged void frac-tions were significantly smaller than the homogenous ones. It isimportant to emphasize that these observations were not con-firmed in a recent work mentioned above (Molla et al., 2011),where the authors experimentally studied bubble train flows in amicrochannel of a rectangular cross-section (50 � 100 lm).Employment of the homogenous void fraction for calculation ofthe bubble length evolution along the channel provided a goodagreement between the calculated and the measured bubblelengths. Moreover, this result is confirmed by the data of Serizawaet al. (2002), who investigated different regimes of the vapor-waterflows in microchannels of small diameters (20, 25 and 100 lm).The measured void fraction values were obtained by image analy-sis of flow patterns observed. The authors reported that even for aslug flow in a channel of the smallest diameter (20 lm), the

measured void fractions were nearly equal to the heterogeneousvoid fractions (see Fig. 10 in Serizawa et al., 2002).

The film thickness di can be evaluated by one of the correlationsknown from literature. Over a wide range of capillary numbers(Ca 6 1), it can be calculated by the semi-empirical equation ofAussillous and Quere (2000):

di

Rc¼ 1:32Ca

23i

1þ 3:33Ca23i

ð10Þ

where Cai = lLUBi/c is the capillary number, c is the gas–liquid inter-facial tension.

Note that at low capillary numbers (the Bretherton’s flow) Eq. (10)is reduced to the theoretical Bretherton equation, valid for Ca < 0.003.Aussillous and Quere (2000) also showed that Eq. (10) is in excellentagreement with the Taylor’s experimental data (Taylor, 1961), whomeasured the bubble velocity UBi and the mean flow velocity Umi ina microchannel flow over wide range of the capillary numbers(0 < Ca < 2). Taylor presented these results as a plot, (UBi� Umi)/UBi

vs. Ca. Aussillous and Quere (2000) demonstrated that the relativefilm thickness di/Rc, calculated from Eq. (7) using the Taylor’s experi-mental data, is very close to that obtained by Eq. (10).

As one can see from Eq. (5), the mass transfer rate from the li-quid to the bubble qRi is the only remaining parameter that weneed to determine to describe the bubble volume evolution alongthe channel.

2.2. Bubble–liquid mass transfer

The mass flux of the dissolved gas from the liquid to the ith bub-ble can be considered as a sum of the following components:

qRi ¼ qsi þ qcbi þ qcfi ð11Þ

where qsi is the gas mass flux from the liquid film to the bubble, qcbi

is the mass flux through the back cap (the left-hand side in Fig. 1),and qcfi is the mass flux through the front cap (the right-hand side inFig. 1).

As mentioned, we assume perfect mixing in each liquid slug;i.e., the dissolved gas concentration in the ith slug is uniform andequal to c0i. For a given slug, the gas concentration differs fromc0i only at the gas–liquid interface where it equals the saturationconcentration csi. Validity of the perfect mixing assumption is dis-cussed below.

2.2.1. Mass transfer from a liquid film to a cylindrical bubble partThe mass flux through a thin liquid film is described by the

convection–diffusion equation. It is more convenient to work withthis equation formulated in a coordinate system moving with thevelocity of a bubble, where the x-axis is oriented opposite the flowabsolute velocity. Because the liquid film thickness is small com-pared to the channel radius, the mass transport across the filmcan be accurately modeled in Cartesian coordinates. This conclu-sion was obtained by Irandoust et al. (1992), who used such an ap-proach for modeling mass transfer through a liquid film in avertical bubble train flow. The y-axis, in this case, is normal tothe interface (y = 0 at the interface). The calculation diagram isshown in Fig. 2. One can see that fluid flows with the bubble

D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155 151

velocity UB through the flat channel. Omitting temporarily assign-ing numbers to the bubbles and slugs, we can write the requiredconvection–diffusion equation as:

@c@x

UB ¼ D@2c@y2 ð12Þ

where c is the dissolved gas concentration in the liquid and D is themolecular diffusivity of the dissolved gas in the liquid.

A regular assumption for the gas–liquid mass transfer model isan equilibrium (saturation) gas concentration on the gas–liquidboundary. This assumption is included into the boundary condi-tions presented below. As mentioned above, the pressure changealong the bubble is practically zero; therefore, the saturation con-centration on the bubble–liquid interface is uniform.

The boundary conditions for Eq. (12) are (see Fig. 2):

c ¼ c0 at x ¼ 0c ¼ cs at y ¼ 0@c@y¼ 0 at y ¼ d

ð13Þ

where c0 is the dissolved gas concentration in the liquid enteringthe clearance between the bubble and the wall and cs is the satura-tion concentration of the dissolved gas, which is a function of thepressure.

The saturation concentration cs on the bubble–liquid interfacecan be calculated by the Henry’s law as a function of the pressure.In the case of a dilute solution, the saturation concentration isdetermined as (e.g., Bird et al., 2002):

cs ¼pH

ð14Þ

where H is the Henry’s constant.Note that at moderate pressure levels, a dependence of the

Henry’s constant on pressure can be neglected (e.g., Bird et al.,2002). For example, our evaluations, performed by using the freelyavailable NIST Chemistry Webbook (webbook.nist.gov/chemistry),demonstrated that this assumption is acceptable for the methane(CH4) – decane (C10H22) system if the pressure does not exceedabout 600 psi (�4.1 MPa).

The analytical solution of the convection–diffusion equation(Eq. (12)) with the boundary conditions, expressed by Eq. (13), is(Souza-Santos, 2007):

cðx; yÞ � cs

c0 � cs¼ 2

pX1n¼1

1� ð�1Þn

nexp � pn

2

� �2 Dx

UBd2

� sin

pn2

yd

h ið15Þ

The corresponding mass flux to the bubble is calculated asfollows:

qs ¼ 2p Rc �d2

� �UB

Z d

0ðc0 � cðLlB; yÞÞdy

¼ 2p Rc �d2

� �UBðc0 � cmðLlBÞÞd ð16Þ

where cmðLlBÞ ¼ ð1=dÞR d

0 cðLlB; yÞdy is the concentration of the dis-solved gas at the clearance outlet averaged over the film thicknessand LlB is the length of the cylindrical bubble portion.

The averaged gas concentration at the clearance outlet is calcu-lated as:

cmðLlBÞ ¼ cs þ ðc0 � csÞ �4p2

X1n¼1

1� ð�1Þn

n2

� exp � pn2

� �2 DLlB

UBd2

� � 1� cos

pn2

� �h ið17Þ

The length of the cylindrical bubble portion is determined bythe following equation:

LlB ¼WB � 4

3 pR3B

pR2B

ð18Þ

2.2.2. Mass transfer through the semi-spherical capsThe mass flux to the semi-spherical cap is calculated as (e.g.,

Bird et al., 2002):

qc ¼ 2pR2BkL;capðc0 � csÞ ð19Þ

where kL,cap is the mass transfer coefficient.The mass transfer coefficient kL,cap can be evaluated using one of

empirical or semi-empirical correlations available in the literature(e.g., Irandoust et al., 1992; Van Baten and Krishna, 2004). In thepresent work, we used the correlation of Van Baten and Krishna(2004), who suggested calculating the mass transfer coefficientfor a cap by an equation obtained on the basis of the Higbie’s pen-etration model, as:

kL;cap ¼2 � 20:5

pDUB

DB

� �0:5

ð20Þ

Because the mass-transfer model, developed by Van Baten andKrishna (2004) for a Taylor flow in a vertical tube, showed an excel-lent agreement with the CFD results, we assume that Eq. (20) issufficiently accurate.

Thus, we determined how to calculate the all mass fluxesthough the total bubble surface (see Eq. (11)). The mass fluxthrough the cylindrical bubble surface is calculated by Eq. (16).The total mass flux through the side caps of the ith bubble is com-puted as follows (see Eq. (19)):

qcbiþqcfi ¼ 2pR2BikLi;capðc0i � csiÞ þ 2pR2

BikLi;capðc0iþ1 � csiÞ

¼ 4pR2BikLi;cap

c0i þ c0iþ1

2� csi

� �ð21Þ

Note that our computations demonstrated that the ratio of themass flux through the cylindrical bubble section to the total massflux through the semispherical caps can be both smaller and largerthan unity. This ratio increases along the micro-channel with an in-crease in the bubble length.

2.3. Liquid slug mass balance

Concentration of the dissolved gas can be calculated by consid-ering mass balance for the liquid slug. Both experimental studiesand CFD simulations demonstrate an intensive mixing within theslug (e.g., Irandoust and Andersson, 1989) caused by strong vortexmotion. Hazel and Heil (2002) modeled the bubble train flownumerically and demonstrated that the vortex motion exists inthe liquid slug if the capillary number does not exceed the criticalvalue Ca = 0.691. In all the computational examples, presented inthis paper, the Ca number is smaller than 0.06. We would like toemphasize again that Van Baten and Krishna (2004) showed thatan engineering model based on the perfect mixing assumption ina slug provides the volumetric mass transfer coefficients, whichare in excellent agreement with those computed using the com-mercial CFD code (CFX). Furthermore, the computations of VanBaten and Krishna (2004) were performed for relatively large cap-illary diameters (Dc = 1.5–3 mm). The weakest contributor to themass transfer across a vortex inside a slug is the diffusion fromthe vortex center to the periphery. According to Bird et al.(2002), the time needed for a system equilibration by the diffusionis proportional to the square of the characteristic system size. Thechannel diameter can serve as a characteristic size for the capillary.Because, our primary interest is in channels smaller than 200 lmin diameter, the diffusion in such capillary flows is much fasterthan in those investigated by Van Baten and Krishna (2004). If

152 D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155

the hypothesis of perfect mixing within the slug is valid for flows inrelatively large channels (Van Baten and Krishna, 2004) it shouldcertainly be valid for microchannels of smaller diameters analyzedin this work. Therefore, we assume that the flow pattern in a slug isperfect mixing.

In the coordinate system moving with the bubble velocity, theslug is immobile and the liquid from the clearance between thebubble and the wall ahead of the slug feeds it. The liquid flowsout of the slug through the clearance behind it with the same flowrate.

The mass balance for the gas dissolved in the ith slug is:

dc0i

dt¼ � q0Ri

Wseið22Þ

where q0Ri is the mass flux from the ith slug to the neighboring bub-bles and Wsei is the effective volume of the ith slug.

The effective volume of the slug is equal to the total liquid vol-ume localized in the pair of ‘‘slug + bubble’’ minus the liquid filmvolume. The liquid film volume increases along the channel withincrease in the bubble length and the film thickness. The effectiveslug volume is calculated as:

Wsei ¼Ws � 2p Rci �di

2

� �diLlBi ð23Þ

The total mass flux from the ith slug to the bubble phase is:

q0Ri ¼ qsi þ qcbi þ qcfi�1 ð24Þ

The mass flux through the i � 1th frontal cap is calculated, bytaking into account Henry’s law (Eq. (14)), as:

qcfi�1 ¼ 2pR2Bi�1kLi�1;capðc0i � csi�1Þ

¼ 2pR2Bi�1kLi�1;cap c0i � csi þ

Dpsli

H

� �� �ð25Þ

where Dpsli is the pressure drop along the ith liquid slugBecause the pressure drop along the ‘‘slug + bubble’’ pair is pri-

marily along the slug, the pressure drop along the slug is evaluatedas:

Dpsli ¼DpL

� �i

ðLB þ LslÞi ð26Þ

where (LB + Lsl)i is the length of the ith ‘‘slug + bubble’’ pair.The length of the ith ‘‘slug + bubble’’ pair is calculated as:

ðLB þ LslÞi ¼ LlBi þWsei þ 4

3 pR3Bi

pR2c

ð27Þ

Because the difference between the radii of the neighboring bub-ble caps and the difference between their mass transfer coefficientshave been shown to be negligible (e.g., Van Baten and Krishna,2004), the equation for the total mass flux from the liquid slug tothe bubble (Eq. (24)), accounting for Eq. (25), takes the form:

q0Ri ¼ qsi þ 2qcbi � 2pR2BikLi;cap

Dpsli

Hð28Þ

Eq. (22), written in terms of the coordinate x, the increment ofwhich was defined above as dx = UBidt, takes the form:

dc0i

dx¼ � q0Ri

WseiUBið29Þ

3. Calculation examples and discussion

The mass transfer model developed is reduced to the integrationof the two ordinary differential equations, describing evolution of

the bubble volume (Eq. (5)) and the concentration of the dissolvedgas in a liquid slug along the channel (Eq. (29)). All other modelparameters are calculated using the algebraic relations presentedabove. Calculations showed that variations of both the dissolvedgas concentration and the saturation concentration within the adja-cent ‘‘bubble + slug’’ pairs are small. Therefore, in the equations forthe ith ‘‘bubble + slug’’ pair, the parameters of neighboring pairs canbe substituted with the parameters for the given pair. By neglectingthe sequential numbering of the bubbles and the slugs, we can writean equation of the gas mass transfer through the side caps, reflectingthe conclusion that the mass fluxes through the front and the backcups are approximately equal, as:

qcbiþqcfi ¼ 2qc ¼ 4pR2BkL;capðc0 � csÞ ð30Þ

The mass flux from the ith slug to the bubble phase can be as-sumed equal to the mass flux to the ith bubble only:

q0Ri ¼ qRi ¼ qR ¼ qs þ 2qc ð31Þ

The mass transfer efficiency in a Taylor flow is usually evaluatedby the volumetric mass transfer coefficient:

kLa ¼ qR

ðc0 � csÞðWs þWbÞð32Þ

Nevertheless, to characterize the closeness of a mixture flowingin a microchanneI to phase equilibrium, we employed the relativedeviation of the actual concentration of the dissolved gas in a li-quid slug from the equilibrium gas concentration:

D ¼ c0 � cs

c0ð33Þ

The criterion D allows for judging how strongly the two-phasesystem is supersaturated at a given channel cross-section; i.e., howclosely the mixture is brought to phase equilibrium. Thus, D char-acterizes the mass transfer device perfectness. In the followingnumerical examples, we will monitor how the criterion D changesalong the microchannel as a function of the channel diameter, thelength, and the bubble generation frequency, and also analyze thecauses of those changes.

It is important to emphasize that though the model developedwas not directly verified by experimental data, its validity is justi-fied. All the model components are either based on the first princi-ples or the verified empirical correlation. Only the two modelassumptions can be considered as not proven directly: (1) perfectmixing in the liquid slug; (2) an empirical correlation for the masstransfer coefficient through the bubble side caps (Eq. (20)). How-ever, as this was explained above, by using these assumptionsVan Baten and Krishna (2004) obtained accurate mass transfercoefficients for channels, whose diameters were order of magni-tude larger than those considered in the current work. Let usemphasize again that these two assumptions are valid. The justifi-cation is as follows:

1. The perfect mixing assumption has to work better for a channelof smaller diameter (see the ‘‘Liquid slug mass balance’’chapter).

2. In a long microchannel (our case), the bubble length is signifi-cantly larger than the bubble diameter over a major part ofthe channel length. The longer the bubble, the smaller contribu-tion of a possible inaccuracy of mass transfer coefficient calcu-lation through the side cap (Eq. (20)) into the overall inaccuracyof calculation of the total mass transfer flux from the bubble.Since Van Baten and Krishna (2004) demonstrated that theircorrelation provides the accurate results for relatively shortbubbles (the bubble lengths exceeded the channel diameteronly by factor �1.6–4), it should be even more accurate for

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

x, m

Volu

met

ric m

ass

trans

fer c

oeffi

cien

t, 1/

s

Dc=25 micrometersDc=50 micrometersDc=100 micrometersDc=200 micrometers

L=1 m

Fig. 4. Distributions of the volumetric mass transfer coefficient along microchan-nels of the different diameters.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

x, m

Rel

ativ

e co

ncen

tratio

n de

viat

ion

from

eq

uilib

rium

, %

Dc=25 micrometersDc=50 micrometersDc=100 micrometersDc=200 micrometers

L= 1 m

Fig. 5. Distributions of the relative deviation of the dissolved gas concentrationfrom its saturation concentration along microchannels of the different diameters.

D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155 153

our case. In the calculations examples, presented below, thebubble length reaches �4 of the channel diameters along initial25–30% of the channel length, exceeding �30 of the diametersat the channel end.

Thus, because the developed model of a flow, accompanied withmass transfer, does not contain non-verified elements able to sig-nificantly reduce its accuracy, it can be reliably used for analysisof microfluidic systems.

For our analysis, we used a two-component mixture of methaneC1 and decane C10 (CH4–C10H22). For all computational examples,we assumed the same fluid parameters. Decane is supersaturatedwith dissolved methane at the initial channel cross-section: theinitial pressure p0 = 430�psi and the temperature T = 298 K whilethe bubble point pressure is pb = 500 psi. The pressure at the mi-cro-channel outlet is pe = 14.7 psi (1 atm). The mixture density isclose to the density of C10 and we assumed that it is constantqm ¼ qL ¼ 700 kg

m3

� �. The liquid viscosity is also assumed to be con-

stant (lL = 5.8 � 10�4 Pa s). The liquid/gas interfacial tension isc = 30 mN/m. The initial concentration of the dissolved methanecorresponds to its saturation concentration at the bubble point(c0 = 14.19 kg/m3). Note that this value was calculated by the freelyavailable NIST Chemistry Webbook (webbook.nist.gov/chemistry).Then the Henry constant is H = 2.43 � 105 m2/s2. The gas constantfor the gaseous methane is R = 518 J/(kg K).

Initially, we investigated how some important flow parametersvary along the channels of the different diameters (see Figs. 3–5).The channel length was fixed at L = 1 m. The initial bubble lengthwas assumed to be half of the channel diameter. The bubble nucle-ation frequency, controlling the initial length of the slug, was as-sumed to be f = 1000 Hz. This frequency was chosen based on theexperimental studies of a bubble train flow in a long rectangularchannel (Molla et al., 2011). The initial slug length Ls0 correspond-ing to the frequency f is estimated as Ls0 = Usl0/f, where Usl0 is themean slug velocity at the initial channel cross-section.

Before moving to the discussion of the modeling results, wewould like explain the reason why in all the figures illustrating thedistributions of flow parameters along the channels, the graphsare broken at different distances ahead of the channel outlet. Theexplanation is as follows. The bubble velocity rapidly increases alongthe channel, especially in the vicinity of the outlet, where a relativepressure reduction is largest. The latter leads to a rapid bubble sizeincrease. In addition, an increase in the bubble velocity (Ca number)leads to liquid film thickening (see Eq. (10)). Increase in the volumeof the liquid film entrained by the bubble leads to a reduction in theadjacent slug size. According to our computational algorithm, the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

x, m

Bubb

le v

eloc

ity, m

/s

Dc=25 micrometers Dc=50 micrometers Dc=100 micrometers Dc=200 micrometers

L=1 m

Fig. 3. Distributions of the bubble velocity along microchannels of the differentdiameters.

calculation procedure is stopped when the neighboring bubblestouch each other. That indicates a transition from a gas–liquid slugflow regime to an annular flow regime, modeling of which is beyondthe scope of the current work. In Fig. 3, we showed the bubble veloc-ity distributions along microchannels of the different diameters(Dc = 25, 50, 100, and 200 lm). The velocity significantly increasestoward the outlet due to the large pressure drop in the channel.One can see that the larger the diameter, the higher the velocity le-vel. This is because the larger the diameter, the lower the frictionlosses; therefore, the larger flow rate is needed to provide the samepressure drop. In Fig. 4, one can see the corresponding distributionsof the volumetric mass transfer coefficient kLa. The smaller is thechannel diameter, the higher the coefficient kLa because the specific(per unit volume) gas–liquid interfacial area, through which themass transport is carried out, rapidly increases with a decrease inthe channel diameter. The coefficient kLa increases along the chan-nel mainly due to an increase in the convective mass transfer fluxwith the increase in the flow velocity. This is important to note thatalthough Ca number increases along the channel due to an increasein the flow velocity, it does not exceed 0.06 even for a flow in thechannel of 200 lm diameter where Ca number level is maximalamong the flows analyzed in this paper. Because the maximum cap-illary number is significantly smaller than the critical capillary num-ber Ca = 0.691 (Hazel and Heil, 2002), limiting an existence of thecounter-rotating vortexes in a slug, all the flows analyzed here are

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

x, m

Rel

ativ

e co

ncen

tratio

n de

viat

ion

from

equi

libriu

m, %

f=100 Hzf=300 Hzf=600 Hzf=1000 Hz

L= 1 mDc=50 micrometers

Fig. 7. Distributions of the relative deviation of the dissolved gas concentrationfrom its saturation concentration along a microchannel for the different bubblegeneration frequencies.

154 D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155

characterized by an intensive (perfect in the model) mixing withinthe slug.

In Fig. 5 we showed how the relative concentration deviation D,selected in our study as a criterion for characterization of the sys-tem deviation from phase equilibrium, changes along microchan-nels of the different diameters. One can see that the criterion Dincreases slightly with an increase in the channel diameter from25 to 50 lm, while a further diameter increase leads to a more sig-nificant increase in the relative concentration deviation. At the ini-tial section of the channel, D rapidly decreases, then reaches a localminimum, after that slowly increases along a relatively lengthychannel section, and after obtaining a maximum drops to almostzero. The decrease in D along the initial channel section is causedby an intensive mass transfer in the system that is far from equilib-rium at the channel inlet. The slow increase in D after passingthrough the local minimum is caused by a slower reduction inthe concentration difference, c0 � cs, in comparison with a decreasein c0 (see Eq. (33)). The steep decrease in the criterion D along thechannel section close to the outlet is explained as follows. As it wasalready mentioned previously, a significant flow acceleration alongthis section is accompanied with the rapid reduction in the effec-tive liquid slug volume, causing an enhancement of the rate ofthe dissolved gas concentration reduction (see Eq. (29)). Thus,the fast reduction in the concentration difference, c0 � cs, leads tothe rapid decrease in the criterion D.

In Fig. 6, one can see how the relative concentration deviation Ddecreases with an increase in the channel length. The profile of Dalong the channel was calculated for microchannels of the constantdiameter (Dc = 50 lm), while the channel length was varied(L = 0.25, 0.5 and 1 m). The bubble generation frequency was main-tained constant (f = 1000 Hz). The results obtained are explained asfollows. The longer the channel, the smaller the flow velocities dueto the fixed pressure drop along the channel. The slower the flow,the longer the contact time between the gas and the liquid phases,which leads to an enhancement in the mass transfer rate per unitlength of the channel. Thus, the overall efficiency of themicro-channel as a mass exchanging device rapidly increases withan increase in the channel length in spite of some reduction ofconvection intensity caused by a decrease in the flow velocity.

Fig. 7 shows the effect of the bubble generation frequency on therelative deviation from the phase equilibrium along a channel of50 lm in diameter and 1 m in length. The calculations were per-formed for the frequencies f = 100, 300, 600, 1000 Hz. The lower thefrequency, the larger the slug length (e.g., f = 100 Hz – Ls0 = 3.2 mm,f = 1000 Hz – Ls0 = 0.34 mm). One can see that the deviations fromequilibrium are relatively large at the low frequencies (long slugs)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

x, m

Rel

ativ

e co

ncen

tratio

n de

viat

ion

from

equi

libriu

m, %

L=0.25 mL=0.50 mL=1.00 m

Dc=50 micrometers

Fig. 6. Distributions of the relative deviation of the dissolved gas concentrationfrom its saturation concentration along microchannels of the different lengths.

and rapidly decrease as the frequency increases. The higher the fre-quency, the shorter the slugs causing the higher gas–liquid interfacialarea per unit slug volume that leads to an increase in the mass transferrate.

4. Conclusions

The model of a bubble train flow accompanied with mass trans-fer through a long microchannel has been developed. In contrast tothe models presented in literature our modeling approach ac-counted for the expansion of gas bubbles and flow velocity in-crease along the channel due to the pressure drop caused byfriction losses. For characterizing mass transfer efficiency, we usedthe volumetric mass transfer coefficient. As a measure of deviationof a gas–liquid mixture from phase equilibrium, we employed therelative deviation of the concentration of a gas dissolved in a liquidslug from the saturation concentration. Using our model, weshowed how the bubble velocity and the mass transfer rate distri-butions vary along the channel in dependence on the channeldiameter. We also studied how distribution of the relative concen-tration deviation along the channel depends on the channel diam-eter, the channel length and the bubble generation frequency. Thesimulation results showed that the mixture flowing in a longmicrochannel of a small diameter is close to phase equilibrium thatcharacterizes such a channel as a very efficient mass exchanger.

Acknowledgment

The authors are thankful to Prof. J.R.A. Pearson (SchlumbergerCambridge Research Center) for the constructive discussion of thiswork.

Appendix A

According to the experimental data and CFD results for micro-channel flows at relatively high Ca numbers (Ca < 0.04), the pres-sure gradient can be calculated as (Kreutzer et al., 2005):

DpL¼ 2qLfU2

m

Dð1� eÞ ¼ 32

D2 lLUmeL 1þ 0:17DLsl

ReCa

� �13

!ðA:1Þ

where Lsl is the slug length, Um = UGS + ULS is the mean flow velocity,UGS ¼ QG

F is the gas superficial velocity, ULS ¼ QLF is the liquid superfi-

cial velocity, F is the area of the channel cross-section, QG and QL are

D. Eskin, F. Mostowfi / International Journal of Heat and Fluid Flow 33 (2012) 147–155 155

the gas and the liquid volume flow rates respectively, and eL = 1 � eis the liquid volume fraction.

Note that the term in parentheses does not depend on thevelocity. Let us consider the following product:

UmeL ¼ ðUGS þ ULSÞeL ¼QG þ QL

Fð1� eÞ ¼ Q L

F¼ ULS ðA:2Þ

The superficial velocity of the incompressible flow in a channelis constant. It then follows from Eq. (A.1) that the mean pressuregradient along the micro-channel, assuming constant liquid viscos-ity and slug length, is also constant.

The slug length may decrease noticeably as a bubble growsalong a channel and a liquid is entrained into the clearance be-tween the bubble and the wall. The slug length evolution may beestimated as:

Lsl ¼Wse

pR2c

¼ Ws

pR2c

�2p Rc � d

2

� �dLlB

pR2c

� Ws

pR2c

� 2dRc

LlB ðA:3Þ

where Rc is the channel radius, Ws is the total (constant) liquid vol-ume localized within a pair bubble + liquid slug, Wse is the effectiveslug volume, equal to Ws minus the liquid film volume, and d is thefilm thickness.

Thus, an effect of the slug length on the pressure gradient can beeasily evaluated. However, the accuracy of the semi-empiricalpressure drop model (Eq. (A.1)) may not be high enough to accu-rately account for this effect. It is important to note that the exper-iments by Molla et al. (2011) showed that the pressure gradient isvery close to constant within a wide range of flow regimes.

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