International Journal of Heat and Mass Transfer 84 (2015) 214–224

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Modeling criteria for extraction regime transitions for microscale in-situvapor extraction applications

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.12.0700017-9310/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 541 737 7017.E-mail address: james.liburdy@oregonstate.edu (J.A. Liburdy).

Saran Salakij, James A. Liburdy ⇑, Deborah V. PenceMechanical Engineering, Oregon State University, Corvallis, OR, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 August 2014Received in revised form 11 November 2014Accepted 25 December 2014

Keywords:Microscale heat sinksVapor extractionFlow map

A major detriment of two-phase microscale flow systems is a significantly large pressure drop. For flowboiling the potential for flow instabilities is also a major concern. Both disadvantages may be suppressedby extracting vapor through a hydrophobic porous wall of the channel as a means to reduce the channelvapor fraction. The vapor extraction may occur as different regimes either as evaporation, bubble extrac-tion or a combination of both. In the design of vapor extraction systems, it is important to accurately pre-dict extraction rates, different extraction regimes, and the effect of extraction on the heat transfer andchannel flow conditions. This study focuses on two parts: the development of physic-based models forthe transition criteria among (i) the extraction mechanism regimes, and (ii) the extraction flow regimesfor microscale flow boiling. The identification and conditions for the various extraction regimes are dis-cussed and criteria for transition are developed based on physical concepts. Six potential extractionmechanism regimes are identified: (a) no extraction, (b) evaporation, (c) bubble extraction, (d) bubbleextraction with partial liquid blockage, (e) bubble extraction with evaporation, and (f) liquid break-through. Based on the criteria for the extraction mechanism regimes, the rate of vapor extraction is mod-eled and used to analyze the effects of vapor extraction on the dynamics of two-phase flow boiling. Theresults show six extraction flow regimes for two-phase flow boiling: (i) single-phase evaporation, (ii)two-phase evaporation – bubble collapse, (iii) full extraction – stable, (iv) full extraction – unstable,(v) partial extraction – stable and (iv) partial extraction – unstable.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Due to the increased development of high performance elec-tronic devices in the last few decades, as well as their miniaturiza-tion, the large increase in heat dissipation per unit volume hasbecome a major concerned. Cooling technologies utilizing two-phase microscale heat sinks have advantages due to high surfaceto volume ratio and large heat transfer coefficients. For two-phaseflow boiling, an additional advantage is the relatively smallstreamwise temperature variation compared with single-phasemethods. However, a major issue of flow boiling in microchannelsis a correspondingly large pressure drop and accompanying flowinstabilities [1,2]. A large pressure drop can lead to large tempera-ture variations along the flow in the two-phase regime. Also, thelarge pressure drop and thermal oscillations due to flow instabili-ties may lead to severe mechanical vibration and dry-out [1].

Numerous investigators [2–10] have developed channel modifi-cations as a means to suppress instabilities, which may be catego-rized into three main groups: (i) use of an inlet restrictor, (ii)application of engineered nucleation sites, and (iii) use of anexpanding channel. The first group [2–4] stabilizes flow boilingby placing a flow restrictor at the inlet to decrease the reversevapor flow. However, the drawback of this method is a largeincrease in the pressure drop of the system [5]. The second group[5–7] fabricates artificial nucleation sites on the channel walls. Thisreduces the required superheat surface temperatures and therebyreduces the rapid expansion of the bubble. The third group [5,8–10] uses an increasing flow cross sectional area along the flowdirection causing bubbles to expand downstream rather thanupstream, leading to less vapor reverse flow.

An alternative approach using in-situ vapor extraction, whichhas shown to potential stabilize flow boiling, has been investigatedby Salakij et al. [11]. The goal of in-situ vapor extraction is tolocally extract the generated vapor from the channel through ahydrophobic porous wall so as to reduce or control the vapor frac-tion inside the channel. In a thermal management application theextracted vapor would most likely be condensed and reused as a

Nomenclature

A areaBo⁄ modified Boiling number, Bo� ¼ q

_milvCav Capillary number, Cav ¼ ll jv

rCb dimensionless parameter for minimum wall superheat

requirementCw rapid wave growth factorcp specific heatdp equivalent membrane pore sizeDh hydraulic diameterFr⁄ modified Froude number, Fr� ¼

ffiffiffiffiffiffiffiffiffiffiqv

ql�qv

qjvffiffiffiffiffiffiffiffiffiffiffiffiffi

gH cos ap

g acceleration due to gravityG mass fluxH channel depthHl liquid level heighth heat transfer coefficienti enthalpyilv heat of vaporizationJa Jacob number, Ja ¼ ql

qv

cp;lDTsub

ilvj superficial velocityk thermal conductivity_m mass flow rate

Nextr extraction number, Nextr ¼ _mextr= _min

Nu Nusselt numberPextr extraction absolute pressure}w wetted perimeterq heat input rateq00 heat fluxS contact surface per unit lengthSt stability parameterT temperatureu velocityv specific volumeW channel widthWe Weber number, We ¼ G2Dh

qlrX Lockhart–Martinelli parameterx thermodynamic equivalent qualityx�out ideal exit quality without vapor extraction,

x�out ¼q

_min�cp;lDTsub;in

ilv

Y dimensionless parameter presented in [25],

Y ¼ ðql�qv Þg sin bðdP=dzÞjv

Greeksa void fractionac channel aspect ratio, ac ¼W=Hb channel inclination angledc surface roughnessdt thermal boundary thickness, dt ¼ kl=hloDPextr extraction pressure differentialDTsat wall superheat, DTsat ¼ Tw � Tsat

DTsub subcooling, DTsub ¼ Tsat � Tbhc contact anglehd half-diverging anglej specific membrane permeabilityl dynamic viscositym kinematic viscosityq densityr surface tension

Subscriptsb bulkbub bubblec cross-sectionalextr extractionheat heatedi liquid–vapor interfacein Inletl liquid phaselo all-liquidmem membraneONB onset of nucleate boilingout outletsat saturationv vapor phasew wall

S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224 215

coolant, allowing for efficient condenser design having only vaporinlet flow. Salakij et al. [12] combined a channel with an expandingcross section with vapor extraction and shows significant improve-ment to the allowable stable heat flux.

Apart from the ability to stabilize the flow, several studies sug-gest that in-situ vapor extraction also has the potential to reducethe system pressure drop while maintaining the benefit ofenhanced heat transfer [13–18]. Apreotesi et al. [13,14] experi-mentally investigated flow boiling of water through a fractal-likebranching microchannel network with in-situ vapor extraction,showing a decrease in the system pressure drop with increasingextraction pressure differential. A later work by Salakij et al. [18],using a one dimensional predictive model validated against theexperimental results obtain in [13,14], shows up to a 70% decreasein the overall pressure drop. Moreover, the bulk fluid temperaturewithin the channel was shown to decrease indicating the potentialof in-situ vapor extraction to decrease the overall operating tem-perature of the device. David et al. [16] investigated two-phaseflow in parallel microchannels with vapor venting, where the flowand venting channels were separated by hydrophobic porousmembrane. Their results show a significant decrease in pressuredrop. The computational model of the vapor-venting process stud-ied by Fang et al. [17] also confirmed this result showing thatvapor-venting helps suppress local dry-out in microchannels.

In order to fully utilize the potential of in-situ vapor extractionfor two-phase flow, it is necessary to understand the effects ofvapor extraction on heat transfer and flow conditions. The effectsof vapor extraction are directly related to the vapor extractionmass flow rate. A number of studies [12–19] have predicted thevapor/gas average velocity, Vextr, transport across the membranebased on Darcy’s law as:

Vextr ¼jlvrPextr ð1Þ

Note that all symbols are identified in the Nomenclature. Thismodel may indeed require added complexities to be accurate forthis application. For example, Salakij et al. [12,18] related vacuummembrane distillation to vapor extraction and included evapora-tion effects on the vapor extraction where the evaporation rate isbased on the local vapor pressure gradient across the membrane.Cappello et al. [20] used the dusty gas model, as a general formof Darcy’s law, with added effects of membrane compaction tosuccessfully predict gas and superheated vapor transport throughthe membrane.

Several studies suggest that two-phase hydrodynamic condi-tions near the membrane affect vapor extraction. Alexander andWang [21] studied vapor separating from a two-phase microscaleflow through a hydrophobic porous plate, called a breather. A

216 S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224

correlation was developed based on the assumption that theextraction rate is related to the flow velocity as a function of theratio of the pressure differential across the porous plate to thepressure drop associated with drag on the bubble. Xu et al. [19]studied gas bubble removal in a microscale channel and proposedcriteria for complete bubble removal by considering film forma-tion, bubble size, and liquid breakthrough conditions. Cappelloet al. [20] studied both air and vapor transport through a porousmembrane from two-phase reservoir, showing that the extractedmass flow rate deviates from single-phase transport results. Theyproposed that these deviations may be caused by the two-phasehydrodynamics near the membrane as well as condensation withinthe membrane.

The development of both flow regime-based pressure drop andheat transfer models depends strongly on the two-phase flowregime [22]. Flow regime transition criteria for microchannels havebeen developed using empirical methods, (e.g. [22–26]). For flowboiling with vapor extraction David et al. [27], using flow visualiza-tion observed two-phase flow regimes in a vapor-venting micro-channel, indicates that stratified flow is a dominated flow regimefor low liquid velocities. With increasing liquid velocities, annularflow occurs. The pioneering work by Taitel and Dukler [28], and thelater work by Barnea et al. [29], for larger flow channels, developeda dimensionless form of basic relationships to predict the condi-tions for flow regime transition.

The goal of the present study is to develop a set of physics-based models for the transition criteria among specified regimesapplicable to in-situ vapor extraction in two-phase flow in micro-channels. Here the flow regimes are identified and the models forextraction are expected to help the development of regime-basedextraction mass flow rate predictions, as well as being usefuldesign tools for enhanced performance of liquid–vapor systems.

2. Physics-based regime map

The methodology to develop a physics-based regime map canbe divided to four main steps: (i) identify possible regimes andphysical conditions for each regime, (ii) develop theoretical modelsfor transition between physical conditions, (iii) nondimensionalizethe transition criteria, and (iv) plot transition criteria on an appro-priate map. The additional step to make a regime map more prac-tical for a specific application is converting the nondimensionalmap into a dimensional map for a specific set of conditions, suchas a working fluid and flow geometry.

The extraction map is used to identify physical conditionsrelated to vapor extraction. The main goal of this map is to helpunderstand the potential of vapor extraction and provide a toolfor design applications. In this study, two main types of extractionmaps are proposed: (i) extraction mechanism map, which charac-terizes how vapor is being extracted, and (ii) extraction flow map,which characterizes the effects of flow boiling on vapor extraction.The former deals with the vapor extraction flow physics throughthe membrane, while the latter defines the two phase channel flowconditions and how this may affect extraction. Rather than devel-oping a single extraction regime map, that includes all consideredextraction regimes, this study focuses on the development of eachtransition criterion and the corresponding individual map for eachspecific regime transition.

2.1. Extraction mechanism regimes

Vapor transport can be initiated by applying a pressure differ-ential across a membrane. For a hydrophobic porous membrane,the liquid phase is suppressed from leaking into the membranepores by surface tension forces, provided the pressure difference

across the membrane is below a critical value. This maximum pres-sure differential across the membrane without liquid leakage isdefined as the breakthrough pressure. By applying Young–Laplaceequation for a straight capillary pore, the breakthrough pressure isestimated as:

DPbreak ¼ �4rlv cos hc;mem

dpð2Þ

It should be noted that although the breakthrough pressurecan be theoretically estimated as in Eq. (2), many researchersexperience breakthrough at much lower pressure differential[16,19,21,27]. This may be because of the complex nature of porousmembranes having a range of pore sizes and shapes.

Vapor extraction can occur in two modes: bubble extractionand evaporation. Bubble extraction is a consequence of hydrody-namic transport through the membrane, i.e. the vapor phase isextracted through a membrane directly by a pressure differentialacross the membrane. Evaporation is a result of thermodynami-cally driven transport where the liquid phase at the membraneevaporates and flows through the membrane via thermal and pres-sure forces. The evaporation occurs when the vapor pressure of theliquid within the channel in contact with the membrane is greaterthan the extraction pressure on the opposite side of the membrane.More detail explanation of evaporative transport mechanism canbe found in vacuum membrane distillation applications, e.g. [30–33]. Both modes of vapor extraction may coexist depending onextraction conditions and whether liquid, vapor or both phasesare in contact with membrane. The physical conditions of vaporextraction and breakthrough are summarized in Table 1.

Extraction mechanism regimes are classified by using mem-brane transport mechanisms coupled with membrane contact con-ditions. There are six potential extraction mechanism regimesidentified as: (a) no extraction, (b) evaporation, (c) bubble extrac-tion, (d) bubble extraction with partial liquid blockage, (e) bubbleextraction with evaporation, and (f) liquid breakthrough. The phys-ical conditions of each regime are summarized in Table 2 wherethe characteristic of each regime are described as below:

(a) No extraction is when the pressure differential across themembrane is not sufficient to initiate either bubble extrac-tion or evaporation.

(b) Evaporation is when the membrane extraction area is solelyin contact with evaporating liquid and the resultant vaporflows through the membrane.

(c) Bubble extraction is when only vapor is in contact with themembrane and the vapor inside the channel is extractedthrough the membrane.

(d) Bubble extraction with partial liquid blockage is when bubblesare extracted when a fraction of the extraction area isblocked by liquid which does not evaporate.

(e) Bubble extraction with evaporation is similar to (d), howeverthe liquid in contact with the membrane also evaporates,i.e. both modes (b) and (c) occur.

(f) Liquid breakthrough occurs when the pressure differenceacross the membrane is greater than the breakthrough pres-sure and liquid flows from the channel through themembrane.

It should be noted that extracting vapor from the channel flowmost likely will result in a change in the phenomenological hydro-dynamic conditions in the channel. As a result, extraction mecha-nism regimes can be seen as local conditions along the flow. Forexample, a high temperature single-phase flow enters a channel,liquid in contact with the membrane may evaporate, where theliquid temperature near the membrane decreases along the flow.

Table 1Mechanism and conditions of membrane transport.

Transport mode Mechanism Phasetransport

Conditions

Evaporation Thermodynamictransport

Vapor � Liquid in contact with hydrophobic membrane� Saturation pressure of liquid at membrane interface is higher than extraction

pressure

Bubbleextraction

Hydrodynamictransport

Vapor � Bubble in contact with hydrophobic membrane� Bubble pressure is higher than extraction pressure

Breakthrough Hydrodynamictransport

Liquid Pressure differential across the membrane is greater than the breakthroughpressure

Table 2Phenomenological conditions of extraction mechanism regimes.

Regimes Phenomenological conditions

Membrane contact phase Transport mechanism

(a) No extraction Liquid or vapor None(b) Evaporation Liquid Evaporation(c) Bubble extraction Vapor Bubble extraction(d) Bubble extraction with partial liquid blockage Both Bubble extraction(e) Bubble extraction with evaporation Both Bubble extraction and evaporation(f) Liquid breakthrough Liquid or partial vapor Breakthrough

S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224 217

This may suppress downstream evaporation, resulting in a changeof regime from pure evaporation to no extraction. Further, a purelystratified flow may change downstream due to extraction into bub-bly flow, resulting in a change from purely bubble extraction tobubble extraction with evaporation.

2.2. Extraction mechanism regime transition criteria

Extraction mechanism regime transition criteria are based onmodeling the transition of the membrane contact phase and mem-brane transport mechanism. The transition criteria can then beused together with the extraction mechanism regime conditions,summarized in Table 2, to identify the regime. Transition criteriaare discussed in detail below.

2.2.1. Membrane contact phaseThere are three possible situations for fluid to be in contact with

the membrane: (i) vapor only, (ii) liquid only and (iii) a mixture ofboth phases. Two criteria are used to separate these three situa-tions: liquid film formation and stratification to an intermittentflow regime. For liquid to be the only phase in contact with themembrane during two phase flow, there must be a liquid film for-mation on the membrane. To separate between vapor only andmixed phases in contact with the membrane, the contact charac-teristics are assumed to be related to the two-phase flow regimeof the fluid inside the channel. In other words, for stratified flowthe vapor phase will generally be the only phase in contact withthe membrane (this assumes extraction occurs at the top of thechannel). Both phases tend to be in contact with the membranefor an intermittent flow. It should be noted that although stratifiedflow is rarely found in most microscale flow boiling applications,stratified flow may likely be found in flow boiling with in-situ

vapor extraction applications. This is because at least one wall ofthe channel is formed by a hydrophobic porous membrane, whichlacks sufficient surface tension forces to pull the liquid to this wall,causing an extended region of stratified flow. The experimentobservations by David et al. [27] support this situation, in thatstratified flow is reported to be the dominated flow in low velocitysituations with sufficient vapor content.

2.2.2. Liquid film formationLiquid film formation is evaluated based on the fact that the

dynamic contact angle decreases with increasing bubble velocity.As the dynamic contact angle approaches zero, the bubble criticalvelocity necessary to form a liquid film at the surface is estimatedbased on criteria given by de Gennes et al. [34]:

ucrit ¼1

9ffiffiffi3p r=ll

ah3

c;mem ð3Þ

where a is a dimensionless parameter estimated to be in the rangeof 15–20. The liquid film forms when the bubble travels faster thanthis critical velocity (ubub > ucrit). For simplicity, the bubble velocitymay be estimated as the vapor superficial velocity, jv. In dimension-less form, this film formation criterion is rewritten in terms of acapillary number, Cav, as:

Cav >1

9ffiffiffi3p

h3c;E

að4Þ

A liquid film formation map of Cav versus hc,E for the range ofvalues of a between 15 and 20, based on Eq. (4), is shown in Fig. 1.

A liquid film, once formed, may rupture for a variety of reasonsbased on the dynamic force balance. However, a simply model canbe presented based on the ratio of the film thickness to surfaceroughness ratio. By assuming annular flow conditions (often found

Fig. 1. Liquid film formation map based on capillary number, Cav, versus membranecontact angle, hc,mem.

218 S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224

in saturated flow boiling in microchannels [35]) with a void frac-tion of a and surface roughness of dc, this criterion can beexpressed as:

a > 1� dc=A1=2c

� �2ð5Þ

The void fraction correlation developed for annular flow by Zivi[36] for a local quality of x can be used:

a ¼ 1þ 1� xx

� �qvql

� �2=3" #�1

ð6Þ

By combining Eqs. (5) and (6), the film rupture map given inFig. 2 shows the film rupture region versus local quality x. A combi-nation of the film formation and rupture maps can be used to esti-mate the existent of liquid film on the membrane. For example, forflow boiling with constant mass flux, a film may form with increas-ing quality due to flow acceleration resulting from phase changealong the flow. As the quality increases the liquid phase may beinsufficient to form a film as predicted from Eqs. (5) and (6).

2.2.3. Stratified to intermittent flow regime transitionThe criterion for two-phase flow regime transition from strati-

fied to intermittent flows is used here to identify the transitionfrom vapor only to two phase contact with the membrane. The cri-terion developed by Taitel and Dukler [28] is used in this study, butis modified for rectangular cross section channels, where the lift,caused by vapor acceleration over a liquid–vapor wave interface,disturbs the interface leading to the rapid growth of interfacialwaves which may then block the vapor flow path. There are twoconditions to be satisfied. First, the lift, due to the decreased pres-sure of the accelerating vapor over the wave interface overcomesthe damping gravitation force. Second, the liquid phase mass frac-tion is sufficient to maintain slug or annular flow.

Fig. 2. Film rupture map based on nondimensional surface roughness, dc/A1=2c ,

versus quality, x.

By considering a finite solitary wave, Taitel and Dukler [28]expressed the condition for rapid wave growth in terms of therequired average vapor velocity as:

uv > Cwðql � qvÞg cos b

qv

Ac;v

dAc;l=dHl

� �1=2

ð7Þ

where Cw ¼ A0c;v=Ac;v , is the ratio of is the cross-sectional area ofvapor flow, Ac;v , and the cross sectional area of the finite wave peakA0c;v , and ranges between 0 and 1. The rapid wave growth conditioncan be rewritten as a function of modified Froude number, Fr�, for ageneral channel geometry as:

Fr� >

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

wAc;v

Ac

� �3 Ac=HdAl=dHl

sð8Þ

where Fr� is defined as:

Fr� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qvql � qv

rjvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gH cos bp ð9Þ

For rectangular channel, Eq. (8) is rewritten as:

Fr� > Cwð1� Hl=HÞ3=2 ð10Þ

The area ratio Cw is estimated by Taitel and Dukler [28] as theratio of the liquid to channel depth:

Cw � 1� Hl=H ð11Þ

Substituting this estimated value of Cw, the criterion for rapidwave growth in a rectangular channel becomes:

Fr� > ð1� Hl=HÞ5=2 ð12Þ

Based on the equilibrium stratified flow condition provided byTaitel and Dukler [28] the relative liquid height is a unique func-tion of the Lockhart–Martinelli parameter X2, denoted as:

X2 ¼ ðdP=dzÞjlðdP=dzÞjv

ð13Þ

In addition, for non-horizontal flows the gravitational influ-ences are represented by the nondimensional parameter Y:

Y ¼ ðql � qvÞg sin bðdP=dzÞjv

ð14Þ

The condition for an equilibrium stratified flow is developed byconsidering the momentum equation of each phase and thenequating the pressure drop terms of each. Using this condition,the criterion for rapid wave growth in circular tube is written asa function of Fr�, X2 and Y. The resultant condition for equilibriumstratified flow as modified for any channel cross sectional geome-try is:

X2 Sl

S

� �nþ1 Ac

Ac;l

� �3" #

� Y � Sv þ Si

S

� �m Ac

Ac;v

� �2 Sv þ Si

S

� �Ac

Ac;v

� ��"

þ Si

S

� �Ac

Ac;l

� ��¼ 0 ð15Þ

The exponents m and n are the exponents of the Reynolds num-ber in the Blasius equation for vapor and liquid phases, respec-tively, which are equal to 1.0 for laminar flow and 0.2 forturbulent flow. Note that for laminar–laminar or turbulent–turbu-lent flow boiling (n = m), X2 can be rewritten as a function of qual-ity, x, as:

X2 ¼ ll

lv

� �n 1� xx

� �2�n ml

mv

� �ð16Þ

Fig. 4. Membrane contact phase map based on Froude number, Fr⁄, versus theLockhart–Martinelli parameter, X2, for horizontal two-phase laminar–laminar flowin a rectangular channel without liquid film formation for a range of channel aspectratios.

S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224 219

The cross-sectional area, Ac is dependent on channel geometryas shown in Fig. 3. For a rectangular channel, S and Ac in Eq. (15)are evaluated as a function of Hl=H and channel

AAl¼ ðHl=HÞ�1 ð17Þ

AAv¼ ð1� Hl=HÞ�1 ð18Þ

Sl

S¼ 2Hl=H þ ac

2ð1þ acÞð19Þ

Sv þ Si

S¼ 1� Hl=H

1þ acð20Þ

Si

S¼ ac

2ð1þ acÞð21Þ

The condition for rapid wave growth in a rectangular channel isevaluated as a function of Fr�, ac , X2 and Y by combining Eqs. (10),(15), (17)–(21).

As previously mentioned, in addition to rapid wave growth, theliquid level must be sufficient to form a liquid slug. Taitel and Duk-ler [28] suggested that the transition between intermittent andannular flow takes place for:

Hl=H > 0:5 ð22Þ

while in the later work by Barnea et al. [33] recommend that thecriterion for transition is:

Hl=H > 0:35 ð23Þ

Using this latter criterion along with the condition for equilib-rium stratified flow, Eqs. (15), (17)–(21), forms a relationshipamong ac, X2 and Y for regime transition.

It should be noted that Barnea et al. [29] suggests that the rapidwave growth and sufficient liquid level conditions, developed byTaitel and Dukler [28], is not sufficient to be applied to small chan-nels. Rather, surface tension forces that attract liquid to the chan-nel walls may need to be included. However, this additionalcriterion is not applied here since the membrane used for vaporextraction application is typically hydrophobic, with minimal sur-face tension forces, so this effect is expected to be minimal.

Based on the above approach, Fig. 4 shows the membrane con-tact phase for horizontal two-phase laminar–laminar flow in arectangular channel without a liquid film formation using liquidlevel criterion from Barnea et al. [37]. The region on the right rep-resents intermittent flow which corresponds to mixed phases incontact with the membrane. The sudden change in slope nearX2 = 1 is due to the change from rapid wave growth to the sufficientliquid level condition. This shows that for low values of X2, i.e. highquality, the criterion is independent of Fr⁄, because the liquid phasevolume is not sufficient to form on the membrane. It also showsthat the vapor only region extends to larger X2 values with increas-ing channel aspect ratio. This is due to the fact that as the channelbecomes wider and there is larger contact area for the vapor phase.

Fig. 3. Cross-section of an equilibrium stratified flow in a (a) rectangular channeland (b) circular tube.

Therefore, the required quality to match the pressure drop of bothphases in the channel is less, i.e. a higher value of X2.

A dimensional membrane contact phase map is developed bycombining conditions for liquid film formation with transitionfrom stratified to intermittent flow. Two examples of this mapare shown in Fig. 5. The lines denoted as A, B, C and D are basedon film formation, Eq. (4), film rupture criteria, Eq. (5), rapid wavegrowth, Eq. (12), and sufficient liquid level, Eq. (23), respectively.Two versions are presented: Fig. 5(a) based on mass flux, G, versusquality, x, and Fig. 5(b) based on superficial liquid and vapor veloc-ities. It is implied by observing Fig. 5(a) that full extraction is hardto achieve since near zero quality there is a ‘‘mixed phases’’ regionwhich potentially reduces the effective bubble extraction area.Since extracting vapor decreases the available quality, the mem-brane contact phase changes from ‘‘vapor only’’ to ‘‘mixed phases’’at around x = 0.015. This supports the observation by several stud-ies [19,20] that vapor extraction from two-phase flow is less effec-tive compared with single-phase membrane transport in someflow conditions. This membrane contact phase map can be usedas a basis to develop models for vapor extraction from two-phaseflow since the vapor extraction should be related to the relativearea of the vapor phase in contact with the membrane.

2.3. Membrane transport mechanism

As previously discussed and summarized in Table 1 the condi-tions to initiate bubble extraction and evaporation are linked tothe following pressure conditions:

Pbub > Pextr ð24Þ

and

PsatjTmem> Pextr ð25Þ

respectively, while the condition for liquid breakthrough is:

Pchan � Pextr > DPbreak ð26Þ

Since in microchannel flows bubble size will generally be on theorder of the channel dimensions the difference in bubble and localchannel pressure is assumed negligible. Therefore, the conditionfor bubble extraction becomes:

Pchan > Pextr ð27Þ

The breakthrough pressure may vary with temperature as itrelies on the surface tension force. To account for this variation,assuming a near linear dependence on surface tension, the

Fig. 5. Examples of dimensional membrane contact phase maps for water at 100 �C and membrane contact angle of 124� in a 500 lm � 500 lm channel with surfaceroughness of 5 lm in terms of (a) x versus G, and (b) jv versus jl; note that the lines A, B, C and D represent transition criteria based on film formation, film rupture, rapid wavegrowth, and sufficient liquid level conditions, respectively.

220 S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224

breakthrough pressure may be written in term of the surface ten-sion ratio and experimental breakthrough pressure at a specifictemperature, Ta, the breakthrough condition becomes:

Pextr > Pchan �rjTmem

rjTa

DPbreakjTað28Þ

Using the conditions for evaporation, bubble extraction, andbreakthrough, Eqs. (25), (27) and (28) the membrane transportmechanism can be determined based on channel pressure, mem-brane temperature and extraction pressure. For example, a mem-brane transport mechanism map for water at 100 kPa-a andmembrane breakthrough pressure differential of 70 kPa at 25 �Cis shown in Fig. 6 as a function of extraction pressure and mem-brane temperature. Lines A, B and C are based on evaporation, bub-ble extraction, and breakthrough conditions, respectively. Notethat it is necessary to consider the membrane contact phase cou-pled with this transport map. For example, although the regionat 75 �C with 50 kPa-a extraction pressure in Fig. 6 is shown as‘‘Evaporation/Bubble extraction’’, only evaporation would occursif a liquid film is formed on the membrane.

2.4. Extraction flow regimes

Vapor extraction affects flow boiling, consequently flow condi-tions change and thereby so does the vapor extraction conditions.In other words, vapor extraction and flow boiling conditions arecoupled. In this section, vapor extraction affected by flow boilingis characterized as the extraction flow regime based on the extrac-tion mechanism regimes (b)–(e) shown in Table 2.

Fig. 6. Membrane transport mechanism map for water at 100 kPa-a and membranebreakthrough pressure of 70 kPa, at 25 �C; lines A, B and C represent transitioncriteria for evaporation, bubble extraction and liquid breakthrough, using Eqs. (25),(24) and (28), respectively.

Six extraction flow regimes are identified by criteria based onthe amount of vapor leaving the outlet of the channel, the boilingcondition, and flow stability. These are: (i) single-phase evapora-tion, (ii) two-phase evaporation – bubble collapse, (iii) full extrac-tion – stable, (iv) full extraction – unstable, (v) partial extraction –stable and (iv) partial extraction – unstable. The phenomenologicalconditions of each regime are summarized in Table 3 where thecharacteristics of each regime are described as:

i. Single-phase evaporation is when there is no vapor genera-tion within the channel, but evaporation occurs throughthe membrane.

ii. Two-phase evaporation – bubble collapse occurs during sub-cooled boiling, and the generated bubbles are condensed inthe flow rather than being extracted, concurrently evapora-tion occurs at the membrane.

iii. Full extraction – stable is when all generated bubbles areextracted and the flow is stable.

iv. Full extraction – unstable is when all generated bubbles areextracted, but the vapor extraction process is not sufficientto stabilize the flow.

v. Partial extraction – stable has some generated bubblesextracted, leaving excess vapor exiting through the channeloutlet while the flow is stability.

vi. Partial extraction – unstable is similar to regime (v) but theflow is unstable.

2.5. Extraction flow regime transition criteria

Similar to the method used to identify the extraction mecha-nism regime transition discussed previously, the transition criteriaof flow dynamics are not developed directly. Instead, the physicalconditions related to extraction flow regimes, which are outletquality, boiling condition and flow stability, are analyzed below.The extraction flow regimes are then identified by matching thephysical conditions with the phenomenological conditions for eachextraction flow regime, as summarized in Table 3.

2.5.1. Outlet qualityThe quality at the channel outlet can be evaluated by consider-

ing conservation relationships applied globally to the entire chan-nel as shown in Fig. 7. Combining conservation of mass and energyand neglecting kinetic energy terms results in:

qþ _minðiin � ioutÞ ¼ _mextrðiextr � ioutÞ ð29Þ

The relationship is approximated by assuming cp;l, il and iv areconstant for the entire channel (neglecting pressure variations)and the enthalpy of the extracted vapor is estimated as the

Table 3Phenomenological conditions of extraction flow regimes.

Regime Phenomenological conditions

Outlet quality Boiling condition Flow stability

i. Single-phase evaporation xout < 0 No boiling Stableii. Two-phase evaporation (bubble collapse) xout < 0 Subcooled boiling Stableiii. Full extraction (stable) xout = 0 Saturated boiling Stableiv. Full extraction (unstable) xout = 0 Saturated boiling Unstablev. Partial extraction (stable) xout > 0 Saturated boiling Stablevi. Partial extraction (unstable) xout > 0 Saturated boiling Unstable

Fig. 7. Schematic of a channel control volume illustrating mass and energytransport.

Fig. 8. Extraction number, Nextr, versus x�out for a range of desired outlet quality, xout.

S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224 221

enthalpy of superheated vapor in the extraction chamber, i.e.iextr � iv . The conservation equation for the condition of inlet sub-cooled liquid becomes:

_minðiout � ilÞ ¼ q� _mincp;lDTsub;in � _mextr½ilv � ðiout � ilÞ� ð30Þ

orðiout � ilÞ

ilv¼

q_min� cp;lDTsub;in

ilv�

_mextr

_min1� ðiout � ilÞ

ilv

ð31Þ

The quality at the channel outlet is:

xout ¼ðiout � ilÞ

ilvð32Þ

or, using Eq. (31) is:

xout ¼

q_min�cp;lDTsub;in

ilv� _mextr

_min

1� _mextr_min

� � ð33Þ

which can be rewritten in terms of an extraction number, Nextr, theratio of the extracted and inlet mass flow rates and x�out , the qualityat the channel outlet that would occur without vapor extraction as:

xout ¼x�out � Nextr

1� Nextrð34Þ

where x�out is:

x�out ¼q_min� cp;lDTsub;in

ilvð35Þ

It should be noted that x�out can be rewritten in term of the mod-ified Boiling number, Bo�in ¼

q_minilv

, and Jacob number,Jain ¼

qlqv

cp;lDTsub;in

ilv, as:

x�out ¼ Bo�in �qvql

� �Jain ð36Þ

The relationship given by Eq. (34) is shown in Fig. 8. As can beseen, the quality at the channel outlet decreases with increasingNextr , since both energy and mass are extracted through the mem-brane. Note that vapor extraction can decrease the quality at theoutlet to be negative, i.e. the outlet is a subcooled liquid, due toexcessive evaporation when the extraction mass flow rate isgreater than generated vapor ðNextr > x�outÞ.

2.5.2. Boiling conditionIncipience of boiling occurs when the wall superheat,

DTsat ¼ Tw � Tsat , is sufficient such that the local saturation pres-sure exceeds the combined static and surface tension inducedpressures. Many studies have found that this wall superheatrequirement depends on the size of active nucleation sites. Forexample, the nucleation criterion developed by Hsu [38] showsa range of active nucleation size as a function of wall superheatas:

frc;min; rc;maxg ¼dt sin hc;w

2ð1þ cos hc;wÞDTsat;ONB

DTsat;ONB þ DTsub;ONB

� �

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 8rTsatð1þ cos hc;wÞðDTsat;ONB þ DTsub;ONBÞ

qv ilvdtðDTsat;ONBÞ2

s" #ð37Þ

where the thermal boundary layer thickness, dt, is estimated as kl=h.Generally, for a system with sufficient active nucleation sites, theminimum required wall superheat for the incipience of boilingcan be simplified to:

DTsat;ONB ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8CbrTsatq00w

qv ilvkl

sð38Þ

where Cb was initially proposed by Hsu [38] for pool boiling to be:

Cb ¼ 1þ cos hc;w ð39Þ

Several investigators have modified the variable Cb, examplesare given in Table 4.

To determine whether boiling occurs within the channel, theincipience of boiling criterion may be written in terms of quality.Consequently, the outlet quality can be used as a measure of theoccurrence of boiling, given in Eq. (34). Using definitions of thewall superheat and subcooling, DTsub ¼ Tsat � Tb, then:

DTsub ¼q00

h� DTsat ð43Þ

For all liquid flow up to the point of the onset of nucleate boil-ing, the all-liquid heat transfer coefficient, hlo, is:

Table 4Variable Cb for boiling incipience models.

Author Model Eq.

Sato and Matsumura [39] (pool boiling) Cb ¼ 1 (40)Ghiaasiaan and Chedester [40] (microchannel)

Cb ¼ 22n0:765 where n ¼ rl jTsat�rl jTw

qlu2in

R� and R� ¼ 2rTsat v lv klq00w ilv

h i1=2 (41)

Kandlikar [41] (microchannel) Cb ¼ 1:1 (42)

Fig. 9. Example of a boiling map for flow in 500 lm � 500 lm channel with lengthof 50 mm and contact angle of 53� where inlet mass flux and subcooling are400 kg/m2 s and 10 �C respectively.

222 S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224

hlo ¼Nulokl

Dhð44Þ

At the inception of boiling Eq. (43) is then rewritten as:

DTsub;ONB ¼q00

hlo� DTsat;ONB ð45Þ

Multiplying this by cp;l

ilv, and introducing the quality in the liquid

region as:

x ¼ i� il

ilv¼ � cp;lDTsub

ilv

� �ð46Þ

the local value of quality at the onset of nucleate boiling is:

xONB ¼cp;l

ilvDTsat;ONB �

q00

hlo

� �ð47Þ

For boiling to occur within the channel, the quality at the chan-nel outlet must be greater than the quality for the onset of nucleateboiling, i.e. xout P xONB. Using the relationship among xout, x�out andNextr, presented in Eq. (34), the boiling condition can be determinedin term of x�out and Nextr. The criteria of the boiling conditions aresummarized in Table 5. An example of the resultant boiling mapusing the minimum wall superheat correlation by Hsu [38] isshown in Fig. 9, indicating the subcooled and saturated boilingregimes.

2.5.3. Flow stabilityThe stability criterion applied to this study is based on whether

vapor experiences reverse back flow into the inlet chamber. Thecriterion based on the model proposed by Salakij et al. [12], whichwas developed for flow boiling in a diverging channel with vaporextraction, is used in this study. The stability parameter, St, thatrepresents the ratio of backward to forward forces acting on theupstream side of the liquid–vapor interface of the expanding bub-ble, is expressed as:

St ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqv

q00wAheat;bub� _m00extr iv Aextr;bubqv ilv

1Ac;1þAc;2

� �2Ac;1

G2inAc;1qlþ r}w;1 cos hd

vuuuut ð52Þ

where subscripts 1 and 2 represent the location of upstream anddownstream sides of an expanding bubble, respectively. The transi-tion from stable to unstable flow occurs when St ¼ 1, and is evalu-ated here at the extreme case where an expanding bubble fills thechannel and just reverses into the inlet chamber such that

Table 5Boiling condition criteria.

Boiling condition Criterion Eq.

No boiling x�out�Nextr

1�Nextr

� �< xONB

(48)

Subcooled boiling xONB 6x�out�Nextr

1�Nextr

� �< 0 (49)

Saturated boiling without excessvapor

xONB 6x�out�Nextr

1�Nextr

� �¼ 0 (50)

Saturated boiling with excess vapor x�out�Nextr

1�Nextr

� �>

xONB; xONB P 00; xONB < 0

�(51)

Aheat;bub ¼ Aheat;chan, Aextr;bub ¼ Aextr;chan, Ac;1 ¼ Ac;in and Ac;2 ¼ Ac;out . Itshould be noted that this model does not account for the instabilitydue to insufficient nucleation sites. Since the half-diverging angle ofthe channel, hd , is usually relatively small, on the order of 1–3�, theterm cos hd may be approximated as 1, and the criterion for stableflow, St < 1, is expressed as:

qvq� _mextriv

qv ilv

1Ac;in þ Ac;out

� �2

Ac;in <G2

inAc;in

qlþ r}w;in

" #ð53Þ

By dividing this by G2inAc;in=qv and using and the inlet hydraulic

diameter, Dh;in ¼4Ac;in}w;in

, Eq. (53) becomes:

q� _mextrivGinilv

1Ac;in þ Ac;out

� �2

<qvqlþ 4qvr

G2inDh;in

" #ð54Þ

which can be rearranging as:

q_minilv

< 1þ Ac;out

Ac;in

� �qvql

1þ 4qlrG2

inDh;in

!" #1=2

þ_mextriv_mextriv

ð55Þ

In dimensionless form, this criterion for stable flow is given interms of the extraction number, Nextr, the modified boiling number,Bo�in, and Weber number, Wein, as:

Bo�in < 1þ Ac;out

Ac;in

� �qvq l

1þ 4Wein

� � 1=2

þ Nextrivilv

ð56Þ

where the Weber number is defined as Wein ¼G2

inDh;inqlr

.The stability map for water at 100 �C is shown in Fig. 10 where

the left hand side of the line is the stable region. Fig. 10(a) showsthat the stability region for a uniform cross-section channel isthe smallest when Wein approaches infinity which represents neg-ligible surface tension effects that suppress the instability com-pared with the inertia forces. This case is chosen to show theeffect of varying cross-sectional area ratio, as shown in Fig. 10(b).In general, the stability region can be extended by increasing thecross-sectional area ratio ðAc;out=Ac;inÞ and Nextr , and decreasing Wein.This supports the results from several investigators [5,8–10] thatexpanding the channel improves flow stability.

Fig. 10. (a) Stability map for water at 100 �C for uniform channel cross section; left of each line is the stable region; (b) stability map for the inlet weber number, Wein

approaching infinity, for varying cross sectional channel flow.

S. Salakij et al. / International Journal of Heat and Mass Transfer 84 (2015) 214–224 223

3. Discussion

Ideally the completed extraction mechanism and extractionflow regime maps should be developed in the form of an indepen-dent single map. However, it may not be practical to combine allindividual criteria together in this case because of the large num-ber of independent variables to be included. For example, to getthe complete single dimensional extraction mechanism map for aspecific fluid flow in a specific channel geometry requires fourindependent variables: either (Pextr, Tmem, G and x) or (Pextr, Tmem,jl and jv). As previously suggested, it would be more practical touse individual maps to identify physical conditions of the regime,and then combine the known predicted conditions to identify theregime.

The regime transition criteria developed in this study are fullypredictive based on physical concepts. The important assumptionsused in the development are listed below:

� Film rupture when film thickness is in the same order as surfaceroughness.� Membrane contact phase is strongly dependent on hydrody-

namics of the flow where stratified flow and intermittent flowsare related to vapor only and mixed phase contact, respectively.� Transition from stratified to intermittent flow is caused by rapid

wave growth due to the decreasing in pressure for vapor flow,and the liquid phase must be sufficient to form a slugs [28].� Membrane hydrophobicity prevents the surface tension force

from pulling liquid to the top wall to form slugs.� Negligible pressure differences occur between pressure inside

the bubble and the surrounding liquid.� For extraction flow analysis, fluid properties are assumed to be

constant through the membrane.� Enthalpy of the extracted vapor is estimated as the enthalpy of

superheated vapor inside the extraction chamber.

Most of the assumptions and models used in this study havebeen validated and shown to be reliable for specific conditions inother types of applications. It is expected that to apply these mod-els with reasonable adaptation to in-situ vapor extraction applica-tion, further modification is required. The models for regimetransition criteria in this work are developed such that they canbe adjusted easily. For example, dimensionless coefficients Cw

and Cb for rapid wave growth, and onset of nucleate boiling criteria,respectively, can be modified to capture other effects.

To validate the assumptions and models for extraction mecha-nism regime transition, flow visualization on both sides of themembrane is required such that the membrane contact phaseand breakthrough can be observed. Because an extraction mecha-nism regime depends on the local condition, the test section should

be small and short such that the quality of flow boiling does notvary significantly within the observed section. The membrane sur-face temperature has to be measureable or, even better, controlla-ble. Also, it is important to be able to have a precise quantitativemeasurement of film thickness that forms on the membrane andchannel walls as well as surface roughness and contact angle ofboth membrane and channel walls. For extraction flow regimetransition, precise measurements of mass flow rate, fluid tempera-ture, pressure and quality at the inlet, outlet and extraction cham-ber are required as well as flow visualization to identify boilingmechanism and flow instability.

4. Conclusions

A systematic development of transition criteria for extractionmechanism and extraction flow regimes has been presented. Allof the transition criteria are developed on a basis which can be eas-ily modified to account for additional effects. Although the valida-tion of the extraction mechanism and extraction flow regimetransition criteria was not performed here, most models andassumptions have been validated in part in other applications.The methodology to validate transition models has been proposedand recommended for follow-on study. Examples of membranecontact phase, possibly extraction mechanism and stability maps,and effect of extraction on outlet quality are shown to be tools toidentify regimes.

Conflict of interest

None declared.

Acknowledgments

The authors would like to acknowledge the financial support ofthis work from Office of Naval Research (N00014-09-1-1079), Dr.Mark Spector, program manager.

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