processes

Review A Review on the Hydrodynamics of Taylor Flow in Microchannels: Experimental and Computational Studies

Amin Etminan * , Yuri S. Muzychka and Kevin Pope

Department of Mechanical Engineering, Faculty of Engineering and Applied Science, Memorial University of Newfoundland (MUN), St. John’s, NL A1B 3X5, Canada; [email protected] (Y.S.M.); [email protected] (K.P.) * Correspondence: [email protected]; Tel.: +1-709-7704416

Abstract: Taylor flow is a strategy-aimed flow to transfer conventional single-phase into a more efficient two-phase flow resulting in an enhanced momentum/heat/mass transfer rate, as well as a multitude of other advantages. To date, Taylor flow has focused on the processes involving gas–liquid and liquid–liquid two-phase systems in microchannels over a wide range of applications in biomedical, pharmaceutical, industrial, and commercial sectors. Appropriately micro-structured design is, therefore, a key consideration for equipment dealing with transport phenomena. This review paper highlights the hydrodynamic aspects of gas–liquid and liquid–liquid two-phase flows in microchannels. It covers state-of-the-art experimental and numerical methods in the literature for analyzing and simulating slug flows in circular and non-circular microchannels. The review’s main objective is to identify the considerable opportunity for further development of microflows and provide suggestions for researchers in the field. Available correlations proposed for the transition of flow patterns are presented. A review of the literature of flow regime, slug length, and pressure drop



 is also carried out.

Citation: Etminan, A.; Muzychka, Keywords: gas–liquid; liquid–liquid; microchannel; Taylor flow; two-phase flow Y.S.; Pope, K. A Review on the Hydrodynamics of Taylor Flow in Microchannels: Experimental and Computational Studies. Processes 1. Introduction 2021, 9, 870. https://doi.org/ 10.3390/pr9050870 Multiphase flow in micro-sized structures refers to a microflow in which two or more distinct phases are recognizable, i.e., a carrying or continuous phase and one or Academic Editor: Gianvito Vilé more dispersed phases. A two-phase flow denotes a combination of two distinct phases, including gas, liquid, and solid particles. Gas–liquid (GL) and liquid–liquid (LL) are two Received: 23 April 2021 common types of multiphase flows in microchannels encountered in various practical Accepted: 9 May 2021 applications, such as biomedical, pharmacological, engineering, and commercial purposes. Published: 15 May 2021 Immiscible LL two-phase flows can also be observed in many industrial applications, where the dispersive liquid flow is introduced as droplets into the carrying liquid flow. Publisher’s Note: MDPI stays neutral Intensified liquid–liquid extraction as a separation process is another application for two- with regard to jurisdictional claims in phase flow in oil and gas industries. The gas bubbles with equivalent diameters of the published maps and institutional affil- microchannel diameter in a GL Taylor flow are surrounded by a thin liquid film of the iations. continuous flow and separated by the liquid slugs [1]. The interaction between the different phases, flow rates, thermophysical properties, and geometrical details of the channel characterizes the flow pattern, liquid film thickness, gas bubble or droplet shape, pressure drop, heat, and mass transfer rates. Knowledge of two-phase flow characteristics enables Copyright: © 2021 by the authors. researchers and designers to optimize channel size and operating conditions. Minimizing Licensee MDPI, Basel, Switzerland. the pressure drop can reduce corrosion and erosion while providing a high heat transfer This article is an open access article rate [1,2]. An essential feature of Taylor flow is the flow pattern in the liquid slug, which distributed under the terms and can also be classified in recirculation flow and bypass flow [2,3]. Therefore, knowledge conditions of the Creative Commons of hydrodynamic properties of Taylor flow is certainly required to understand transport Attribution (CC BY) license (https:// phenomena characteristics for improving the performance of microchannels and enabling creativecommons.org/licenses/by/ operational conditions to be optimized. 4.0/).

Processes 2021, 9, 870. https://doi.org/10.3390/pr9050870 https://www.mdpi.com/journal/processes Processes 2021, 9, 870 2 of 40

A review on the hydrodynamic characteristics of Taylor flow was conducted by Angeli and Gavriilidis [1] for circular and non-circular small channels. Correlations for film thickness measurements were summarized and the key effects of Capillary number (Ca) on film thickness were discussed in detail. Recently, Etminan et al. [4] presented all of the correlations proposed for film thickness measuring in the literature. They found the best-fitted correlation between some experimental data and a wide range of Ca. It can be concluded that there is still a lack of certainties for the gradient of surface tension at the interfacial region resulting in discrepancies between experimental predictions and numerical/theoretical findings. Moreover, more attempts should be devoted to finding specific flow and operational conditions for enhancing the performance of microchannels. For multiphase flows, it is necessary to predict the transport phenomena in terms of flow parameters, i.e., superficial flow velocity components, slug length, liquid film thickness, and the flow rates of each phase. Multiphase flow is valid as long as the separation between phases is recognizable at a scale greater than the molecular level. The ability to describe the behavior of multiphase flow requires a set of basic definitions and assumptions. The following subsections have been designed to represent the importance of these basic definitions and the role of dimensionless groups in describing the two-phase flows’ behavior. This paper is aimed to review two-phase flows in microchannels, with a particular focus on the hydrodynamic aspects. Fundamentals of multiphase flows, such as basic definitions and dimensionless parameters are studied. Identification of flow pattern and bubble/droplet formation are discussed in experimental investigations, numerical simulations, and flow regime transitions. Correlations of slug lengths and pressure drop are presented and classified in terms of cross-sectional areas of microchannels.

2. Basic Definitions in Two-Phase Flows The capillary length has been used to identify the compaction in micro-sized applica- tions, such as microchannels in heat exchangers. A micro-sized channel can be adequately assumed when the hydraulic diameter of the channel is less than the capillary length. In contrast, for large-diameter channels, the Laplace number (La) can be properly considered as a suitable length scale for calculations instead of the bubble or droplet diameter. Many other definitions have been developed to describe the characteristics of multiphase flows, including fluid flow, and heat and mass transfers, which are presented in Table1. It should be noted that due to the presence of more than one phase and the interaction between all phases involved in multiphase flow, the thermophysical properties have been expressed differently compared to those of a single-phase flow [3–5].

Table 1. Important definitions for multiphase flows.

Name Symbol Definition Description . . . The sum of mass flow rate of the liquid and the Total mass flow rate m m + m t 1 g gas phases The sum of volumetric flow rate of the liquid Total volumetric flow rate Q Q +Q t l g and gas phases . The total mass flow rate by cross-sectional area Total mass flux G m /A t t of the tube  0.5 The ratio between interfacial and gravitational Capillary length L σ ca (buoyancy) effects g(ρl−ρg) The ratio of average real velocity of the gas and Slip ratio S U /U g l liquid phases The ratio of volumetric flow rate of the gas phase Average velocity of gas phase U Qg Qg Vg to tube cross-sectional area occupied by the gas g A = αA = α g phase flow Processes 2021, 9, 870 3 of 40

Table 1. Cont.

Name Symbol Definition Description The ratio of volumetric flow rate of the liquid Average velocity of liquid Ql Ql Vl Ul = = phase to tube cross-sectional area occupied by phase Al (1−α)A (1−α) the liquid phase flow The velocity of the gas phase if it flows alone in Superficial velocity of gas the tube or the ratio of the volumetric flow rate V Q /A phase g g of the gas phase and the cross-sectional area of the tube The velocity of the liquid phase if it flows alone Superficial velocity of liquid in the tube or the ratio of the volumetric flow V Q /A phase l l rate of the liquid phase and the cross-sectional area of the tube The sum of the superficial velocities of two Mixture velocity V Qt = V +V m A l g phases . . The ratio of the mass flow rate of the gas phase Quality or dryness fraction x m /m g t to the total mass flow rate The ratio of the tube cross-sectional area (or Void fraction α Ag/A volume) occupied by the gas phase to the tube cross-sectional area (or volume) Volumetric quality The ratio of the volumetric flow rate of the gas β Q /Q (dynamic holdup) g t phase to the total volumetric flow rate  0.5 A function of the Lockhart–Martinelli parameter Two-phase friction multiplier ϕ 1+ C + 1 X X2 (X) and the Chisholm constant (C)

3. Dimensionless Parameters A dimensionless number can represent the ratio of two different forces or physical quantities, which play a significant role in the flow pattern and interaction between phases (see Table2). The length scale of a channel, which appears for the majority of dimensionless numbers, is often defined as the diameter of tubes or the hydraulic diameter of ducts.

Table 2. Some of the most common non-dimensional numbers in multiphase flow.

Name Symbol Definition Description ρ ρ −ρ 3 Archimedes Ar l( l g)gd The ratio of the gravitational to the viscous effects 2 µl Bond or Bo 2 ρ −ρ gd ( l g) The ratio of the gravitational (buoyancy) and the capillary force scales Eötvös Eo 4σ The ratio of the interface width and the tube diameter or any other Cahn Cn ξ/d length scale Capillary Ca µU/σ The ratio of the viscous forces and the capillary forces Ca/Re Ca/Re µ2/(ρdσ) (N/A) p Froude Fr U/ gd The ratio between the flow inertia and the external field  0.5 Laplace La σ The ratio of the capillary and the gravitational (buoyancy) effects 2 gd (ρl−ρg) √ The ratio of the viscous force to the inertia and the surface tension Ohnesorge Oh We = √ µ Re σρd forces Reynolds Re ρUd/µ The ratio between the inertia and the viscous forces 2 Suratman Su Re = 1 = σρd The ratio of the surface tension to the viscous forces We Oh2 µ2 ρU2d Weber We CaRe = σ The ratio of the inertial forces to the interfacial forces

The relation of gravitational to viscous forces is Archimedes number (Ar) that the formal definition is given in Table2. This number frequently appears in tube-shaped chemical process reactor designs, which describe the motion of fluids caused by the Processes 2021, 9, 870 4 of 40

difference in densities. The Ar was linked to the Reynolds number (Re) as an explicit- iterative correlation [6]: Ar  −1.214 Re = 1+0.579 Ar0.412 (1) 18 Hua et al. [7] found that ellipsoidal gas bubbles were produced as the Ar and Bond (Bo) numbers were gradually increased. In solid particle multiphase flows, the Ar has also been related to the drag force coefficient and the Re [8–10] and the wind threshold velocities of particles in GL flow [11]. The Ar number was also correlated with the inverse viscosity number (N) by Quan [12] in an upward or downward co-current Taylor flow,

0.5    3  ρl ρl − ρg gd = 0.5 = N Ar  2  (2) µl

For a large N, the elongated tail of the bubbles oscillated for upward flow, and the shortened tail was only observed for downward flow. The Bond number (Bo), also called Eötvös number (Eo), is the inverse of the Ar number and denotes the ratio between the gravitational and capillary effects. Since the Bond number varies with the square of length–scale, each change, even small, in channel diameter causes significant variation in the Bo. For air–water two-phase flow in channels of a diameter less than 1 mm, which are very common in micro-sized applications, the Bo number is approximately in the order of 0.1 and the importance of viscous effects is predominant. Therefore, the gravitational effects can be negligible in most GL flows in microchannels. Hua et al. [7] and Uno and Kintner [13] showed that the gas bubble velocity was significantly impacted by the wall effects for high Bond numbers. They showed that bubble shape was significantly affected by the Bo number: elongated bubbles with spherical caps at both ends appeared for Bo > 50. Some other specific regions of Bo 0.5 and BoRel numbers have been proposed by investigators to categorize the behavior of multiphase flows, for example, dominant surface tension for Bo ≤ 1.5, non-negligible inertia, surface tension, and viscous effects for 1.5 < Bo ≤ 11, and negligible surface tension effects for Bo > 11 [14–17]. The absorption of CO2 in micro-scale reactors where the Chisholm parameter was correlated with the Bo number was studied by Ganapathy et al. [18]. Prajapati and Bhandari [19] quantified the instability increase of boiling flow in a microchannel for the region of Bo < 1. The Cahn number (Cn) is defined as the ratio of the interface width (ξ) and the tube diameter or any other length scale (d),

Cn = ξ/d (3)

Cahn and Hilliard [20,21] and Cahn [22] focused their studies on the free energy of the volume of an isotropic system, where the densities of components involved are non- uniform. The concept of the Cahn number was introduced by them when they found an increase in the thickness with temperature. Soon after, the efforts to obtain the governing equation in a non-equilibrium situation led to a well-known Cahn–Hilliard equation involving the Cahn number for the first time [23–26]. For example, Choi and Anderson [27] coupled the Cahn–Hilliard theory with the Extended Finite Volume Method (XFVM) for the dynamic modeling of suspended particles in two-phase flows. The importance of viscous and capillary effects in two-phase flow in microchannels has been correlated by the capillary number based on the liquid slug velocity,

Ca = µU/σ (4)

The capillary number is a dimensionless group to explain how viscous and surface tension forces affect the interface between the gas and liquid phases, and also between two immiscible liquids. Although, the length-scale does not appear in the capillary number, Processes 2021, 9, 870 5 of 40

surface tension forces become more significant relative to gravity as the cross-sectional area of the channel is decreased. For most microflow applications, the approximate values of superficial velocity range from 10 µms−1 to 1 ms−1, when the viscosity is 10−3 kg m−1 s−1, and the surface tension can be reasonably considered at 0.05 N m−1. These flow conditions cause the Ca to range from 2 × 10−7 to 2 × 10−2. For high viscosity liquids, such as silicone oil at a mixture velocity of 1 ms−1, the Ca becomes ~1. The Ca frequently appears in correlations measuring liquid film thickness [28–52], pressure drop [30,35,44,48,53–57], and slug length [58–61] in multiphase flows. Recently, the poor accuracy of classical pressure drop correlations with an increase of the Ca was emphasized by Ni et al. [51]. This was due to the presence of waves around the tail of the bubble, change of the semi-spherical head of the bubble, and powerful circulation inside the liquid slugs. Patel et al. [52] indicated that the film thickness at the corner of a square microchannel decreased with an increase of Ca, where a linear expression was derived for the film thickness in terms of Ca. Several other researchers showed an increasing trend in film thickness with Ca. These hydrodynamic features will be discussed in the following sections. The Re expresses the ratio of inertial to viscous effects and predicts the flow pattern in different situations: Re = ρUd/µ (5) where ρ, µ, and U are the density, dynamic viscosity of the continuous phase fluid, and the velocity of flow, respectively. The thermophysical properties of a multiphase flow are different from a single-phase flow, which has been discussed by Awad [5] in detail. Regarding the value of the inertia forces, the flow regime remained laminar for the Re numbers less than ~2300 [62,63]. Since the cross-sectional area of a microchannel often has a diameter of 1 mm or smaller and the liquid velocity is less than 1 m s−1, the Re for water as the continuous phase is ~1000 or less. This means that the flow regime in the microchannel applications is normally laminar and the predominant effects of the viscous forces must be taken into account. Because more than one phase is involved, different approaches have been used by researchers to determine the value of Re and the thermophysical properties in the multiphase flows. To realize the importance of the Re definition, consider two following cases: first, the inertia effects of the gas phase are insignificant, when the gas and the liquid superficial velocities are in the same order. For instance, the air to water density ratio in two-phase flow is around 1.225 × 10−3, which indicates the dominant role of the liquid phase density in defining the Re. This situation can be recognized in a bubbly flow pattern. Second, the inertia effects of both phases are important when the gas superficial velocity is relatively high compared with the liquid superficial velocity. The friction factor is necessarily calculated for the pressure drop calculation, while the Re appears in other correlations of hydrodynamic concepts, see Section6 for further information. The Re is in the correlations for liquid film thickness and pressure drop obtained by Heiszwolf et al. [39] and Han and Shikazono [46]. The product of the Re and the We numbers was introduced as a transition criterion from slug to slug-bubbly regimes by Suo and Griffith [64]. Jayawardena et al. [65] proposed the ratio of liquid phase to gas phase the Re numbers to identify the transitions from bubbly to slug and from slug to annular patterns. Several other non-dimensional numbers in multiphase flows have been gathered and discussed by Awad [5]. As the diameter of the channel and the velocity of flow decrease, the inertial, and gravitational effects become negligible, while the surface tension becomes a predominant factor. The knowledge of the hydrodynamics of multiphase flow is strictly intertwined with the dimensionless numbers presented in Table3. According to this table, an accurate description of the flow map, bubble or droplet formation, and transition from one flow regime to another are highly depended on Bo or Eo, Ca, Re, We, Su, and Fr. While, Bo, Ca, and Re have appeared in the pressure drop correlations so far, the friction factor is only governed by the Re. The liquid film thickness has been correlated or described with Re, Ca, and We numbers. Finally, the slug length has only included the Re and the Ca numbers. Processes 2021, 9, x FOR ProcessesPEER REVIEW 2021, 9 , x FOR ProcessesPEER REVIEW 2021, 9 , x FOR PEER REVIEW 6 of 45 6 of 45 6 of 45

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DependencyTable of hydrodynamic 3. aspect DependencyTable of shydrodynamic and 3. aspectDependency theTable of dimensionless hydrodynamics and 3. aspectDependency the of dimensionlesshydrodynamics and numbers. aspect the of dimensionlesshydrodynamics and numbers. aspect the dimensionlesss and numbers. aspect the dimensionlesss and numbers. the dimensionless numbers. numbers. pressurepressure drop droppressurefilmslug thickness length drop Hydrodynamic pressurefilm slug thickness length drop DimensionlessAspects Dimensionlessslug Numberslength Bo ( EoDimensionless ) Numbers CaDimensionless Numbers FrDimensionless Numbers Re Dimensionless NumbersSu NumbersWe Hydrodynamicslugslug lengthHydrodynamic length Aspectspressurefriction slugHydrodynamic length factor drop Aspects Hydrodynamic Bo ( Eopressure filmfrictionAspectsslug) thicknessHydrodynamic length Bo factor drop Ca( EoDimensionlessAspects ) Hydrodynamic Bo ( Fr EofrictionCaAspects Dimensionless ) Numbers Bo factor Re ( EoCa FrAspects Dimensionless ) BoNumbersSu Re( EoCa FrDimensionless ) We BoNumbers Su Re( EoCa Fr Dimensionless ) NumbersWeSu ReCa Fr Dimensionless NumbersWeSuRe Fr NumbersWeSu Re WeSu We flow map, slug profileflow map, slug profileflow map, slug profile Hydrodynamicfilmfrictionfriction thicknessHydrodynamic factor factor Aspectsfilmfrictionslug thicknessHydrodynamic length factor filmAspects thicknessHydrodynamic Bo ( Eopressure filmfrictionAspectsslug) thicknessHydrodynamic length Bo factor drop Ca( filmEoAspects ) thicknessHydrodynamic Bo ( Fr filmEoCaAspects ) thickness Bo Re ( EoCa FrAspects ) BoSu Re( EoCa Fr ) We Bo Su Re ( EoCa Fr ) WeSu Re Ca Fr WeSuRe Fr WeSu Re WeSu We flow map, slugflow profile regimemap, slug transition profileflow regimemap, slug transition profileflow regime transition flowpressurefilm map, thickness slug droppressurefilmfriction profile thickness factor droppressurefilm thickness drop pressurefilmfriction slug thickness length factor drop pressurefilm thickness drop pressurefilm thickness drop flowflowpressure regimeslug regime lengthflow transition drop transitionpressure regimemap, slug lengthslug transition droppressure profileslug lengthflow drop pressurefriction regimemap,slug lengthslug factortransition drop pressure profile slug length drop pressure slug length drop frictionslug lengthflow factor friction regime slug length factortransitionfrictionslug lengthflow factor friction regimemap,slug lengthslug factortransition friction profile slug length factor friction slug length factor flowA graphicalfriction map, slugflow factor flow profilefriction map, map, flowslug factor whichfriction profilemap, flowslugdisplays factor friction regimeprofilemap, flowslug the factortransition friction specific profilemap, flowslug factor region friction profilemap, forslug factor each profile flow regime along with their transition lines, uses a few important characteristics of multiphase flow, such as flow regimemap, slugflow transition profile regimemap, flowslug transition regimeprofilemap, flowslug transition regimeprofilemap, flowslug transition regimeprofilemap, flowslug transition regimeprofilemap, slug transition profile superficial velocity, Re, Ca, and We. Table4 summarizes the non-dimensional numbers andflow other regime flowflow parameterstransition regimeflow transition that regime useflow transition the regime x- andflow transition y-axisregime flow coordinatestransition regime transition for quantitative-based flow map diagrams.

Table 4. Some non-dimensional numbers and thermophysical properties selected by authors for displaying flow maps.

Dispersed Liquid or Gas Phase, #2 2 V Re Ca Re2/Re1 We We Oh Ql/(Ql + Qg) ρV G/λ X Quality V (1) Re (13) Ca (11) (15) (6) Re/Ca (3) We (15) (2) (14) Su (9) (10) We Oh (14) (4) ρV2 (7) Bλψ/G (5) Flow Rate (8) Continuous Liquid Phase, #1 Force (12) The list of authors: (1) Ganapathy et al. [18], Patel et al. [52], Hewitt and Roberts [66], Fukano and Kariyasaki [67], Triplett et al. [68], Zhao and Bi [69], Akbar et al. [70], Kawaji and Chung [71], Cubaud and Ho [72], Günther et al. [73], Yue et al. [74], Kirpalani et al. [75], Yue et al. [76], Dessimoz et al. [77], Roudet et al. [78], Deendarlianto et al. [79], Wu and Sundén [80], Farokhpoor et al. [81] (2) Akbar et al. [70], Yue et al. [76], Zhao et al. [82], Yagodnitsyna et al. [83], (3) Jayawardena et al. [65], Dessimoz et al. [77], (4) Wu and Sundén [80], Yagodnitsyna et al. [83], (5) Baker [84], (6) Suo and Griffith [64], (7) Hewitt and Roberts [66], (8) Sato et al. [85], (9) Jayawardena et al. [65], (10) Jayawardena et al. [65], (11) Cubaud and Mason [86], (12) Kirpalani et al. [75], (13) Dessimoz et al. [77], (14) Yagodnitsyna et al. [83], (15) Wu and Sundén [80].

As a guide to reading this table; the kinetic energy of flow (ρV2) was used by the author’s group of 7 which its row and column are colored red. The first effort to display a flow pattern diagram dates back to 1953 when Baker [84] computed the mass velocity of each phase and two other expressions involving the viscosity and densities (G/λ and Bλψ/G) of both phases (see ‘5’ in Table4). These dimensional terms were not chosen by others afterward. In contrast, it is the interfacial velocities of the phases that have frequently been selected by scholars for more than half a century (see ‘1’ in Table4). After velocity, it seems that Re, We, and Ca numbers have the highest use by researchers for showing the flow pattern clearly and more understandably. Processes 2021, 9, 870 7 of 40

4. Identification of Flow Patterns and Bubble/Slug Formation The first step of transport phenomena analysis of multiphase flows is to explain the flow characteristics in a phase-mapped diagram. The transition lines in such maps specify the criterion of a transition from one flow regime to another. A two-phase flow usually consists of a train of dispersed bubbles or droplets carried by a continuous phase, which po- tentially enhances the rates of heat and mass transfers in industrial applications. Although the flow patterns in microchannels and large-diameter channels show similarities, there are several differences. As the channel diameter increases, the laminar flow regime starts to be unstable due to the inertial effects. Kawaji and Chung [71] and Akbar et al. [70,87] indicated flow patterns in micro and minichannels, where the stratified flow regime was not present for larger channels. The effects of increasing the Reynolds number on the bubble profile were numerically and experimentally discussed by Kreutzer et al. [88]. Their results indicated a more elongated nose and flattened tail due to the lack of large enough interfacial forces to keep the hemispherical shape of the bubble caps as Reynolds number increased. Figure1 presents a schematic of GL flow patterns in capillaries. The geometrical details of the channel, configuration of the channel, properties of phases, superficial velocities of phases, and types of junctions are the predominant factors to determine flow patterns in microchannels. The wetting or drainage effects of each phase are responsible for at least part of the flow maps in multiphase flows. In a bubbly flow regime, the individual bubbles move through the continuous liquid phase at very low liquid superficial velocities. When the bubble size is small, the interaction between bubbles is negligible. Conversely, increasing the gas-to-liquid volumetric flow rate causes the bubbles to coalesce and a breakup occurs (Figure1). Bubble coalescence becomes stronger as the gas-to-liquid volumetric flow rate is increased, to make large long bubbles with a rounded front that spans the cross-sectional area of the channel. Characteristics of the bullet-shaped bubbles most often called Taylor or slug flow in which the Taylor bubbles are surrounded by a thin layer of the carrier liquid phase, Figure1. Some small dispersed bubbles are between the Taylor bubbles as the gas-to-liquid volumetric flow rate is enhanced to make a transition from slug to churn patterns (Figure1e). In a churn flow pattern, neither dispersed nor carrier phases are continuous and irregular plugs of gas flow through the liquid phase and a wide variety of bubble lengths can be observed. This chaotic flow pattern occurs when the velocity is increased (Figure1f). In large-diameter tubes, an oscillating chaotic flow is much more remarkable than that in small-diameter tubes. This flow regime is also named a semi-annular or unstable-slug flow pattern. In an annular flow pattern, a very thin thickness of the liquid phase remains on the walls of the channel for two flow conditions: the gas-to-liquid volumetric flow rate increases and the velocities of phases rise at low liquid volume fraction (Figure1g). While very small-diameter droplets of the liquid phase are in the core of the gas phase, the flow pattern is mist (Figure1h). An experimental study of a hydrophobic ionic/water two-phase flow in two T and Y inlet junctions on the flow patterns was carried out by Tsaoulidis et al. [89], resulting in a negligible impact of the inlet configuration. They also investigated the effects of capillaries’ materials made of glass, Fluorinated Ethylene Propylene (FEP), and Tefzel on flow patterns illustrated in Figure2. Although the boundaries of the FEP and Tefzel are similar, the plug flow regime occupies a larger area in the flow map compared to the FEP capillary, and an annular pattern is observed in higher mixture velocities and lower ionic volume fraction. They did not observe annular and drop patterns in the glass capillary when the continuous phase was water. Processes 2021, 9, 870 Processes 2021, 9, x FOR PEER REVIEW 9 of 45 8 of 40

FigureFigure 1. The schematic 1. The schematicof observed flow of regimes observed in the flowvertical regimes microchannels; in the (a,b vertical) bubbly flow, microchannels; (c,d) Taylor or seg- (a,b) bubbly flow, mented flow, (e) transitional flow from slug to churn, (f) churn flow, (g) annular-film flow, (h) mist-annular or wispy- annular(c,d flow) Taylor [44]. Reproduced or segmented by permission flow, from (e) transitionalKreutzer, M.T.; flowKapteijn, from F.; Moulijn, slug to J.A.; churn, Heiszwolf, (f) churnJ.J., Chemical flow, (g) annular-film Processes 2021, 9, x FOR PEER REVIEWEngineeringflow, (Science;h) mist-annular published by Elsevier, or wispy-annular 2005. flow [44]. Reproduced by permission from Kreutzer,10 of 45 M.T.; Kapteijn, F.; Moulijn,An J.A.; experimental Heiszwolf, study J.J., of a Chemical hydrophobic Engineering ionic/water two-phase Science; flow published in two T and by Y Elsevier, 2005. inlet junctions on the flow patterns was carried out by Tsaoulidis et al. [89], resulting in a negligible impact of the inlet configuration. They also investigated the effects of capillar- ies’ materials made of glass, Fluorinated Ethylene Propylene (FEP), and Tefzel on flow patterns illustrated in Figure 2. Although the boundaries of the FEP and Tefzel are similar, the plug flow regime occupies a larger area in the flow map compared to the FEP capillary, and an annular pattern is observed in higher mixture velocities and lower ionic volume fraction. They did not observe annular and drop patterns in the glass capillary when the continuous phase was water.

FigureFigure 2.2. FlowFlow map boundaries boundaries observed observed in inthree three diffe differentrent glass, glass, FEP, FEP, and andTefzel Tefzel microchannels microchannels with with a T-junction [89]. Reproduced by permission from Tsaoulidis, D.; Dore, V.; Angeli, P.; a T-junction [89]. Reproduced by permission from Tsaoulidis, D.; Dore, V.; Angeli, P.; Plechkova, N.V.; Plechkova, N.V.; Seddon, K.R., International Journal of Multiphase Flow; published by Elsevier, Seddon,2013. K.R., International Journal of Multiphase Flow; published by Elsevier, 2013. 4.1. Experimental Investigations 4.1. Experimental Investigations Transport phenomena for multiphase flows in capillary passages involve the behavior Transport phenomena for multiphase flows in capillary passages involve the behav- ofior each of each phase phase individually, individually, the the mutual mutual interactions interactions ofof phases,phases, and and the the interactions interactions be- between phasestween phases and solid and boundaries.solid boundaries. The The presence presence of ofgas gas bubbles in in a amultiphase multiphase flow flow can can be consideredbe considered as anas an obstacle obstacle to to the the liquid liquid phasephase flow,flow, which is is known known as as the the Jamin Jamin effect effect [90]. This[90]. phenomenonThis phenomenon is prominentis prominent enough enough to to decrease decrease the the productions productions of theof petroleum the petroleum industryindustry duedue to the the high-pressure high-pressure drop drop and and the theflow flow pattern pattern transition. transition. At the Atinterface the interface betweenbetween twotwo componentscomponents of of a a GL GL two-phase two-phase flow, flow, the the surface surface tension tension gradient gradient is substan- is substantial, tial, which affects the interaction between phases and other transport phenomena. Histor- ically, this effect is known as the Marangoni effect, which pushes the liquid phase to flow away from low surface tension regions and was first explained by Gibbs [91]. A comprehensive experimental study on the flow maps by means of high-speed X- ray photography and flash methods simultaneously was reported by Hewitt and Roberts [66], which recognized the previous correlations for each flow pattern proposed by Baker [84] and Wallis [92]. A rough sketch of streamlines in the slug region of a GL flow was experimentally obtained by Taylor [31], where a stagnation point on the vortex and a stag- nation ring on the curvature of the bubble were observed for two extreme cases of velocity fraction, (Vg − Vm)/Vg, greater and less than 0.5. The high-speed photography was com- pared with X-ray technology to provide clear images of air–water flow regimes through a capillary [66]. Irandoust and Andersson [34] experimentally depicted Taylor’s bubble pro- file, where the film thickness remained constant in the middle of the gas bubble. They also confirmed that the gas bubble was elongated in the front and squeezed in the rear parts. A review study on the flow maps was conducted by Akbar et al. [70] to show the dominant effects in each flow regime using available experimental data. They proposed a hydraulic diameter of 1 mm as a criterion for classifying the previous studies into the five regions: bubbly, plug or slug (surface tension dominated), annular (inertia dominated), dispersed, and transition zones regarding the amounts of superficial velocities and the Weber num- bers. Micro-particle image velocimetry (µ-PIV) and fluorescence microscopy techniques

Processes 2021, 9, 870 9 of 40

which affects the interaction between phases and other transport phenomena. Historically, this effect is known as the Marangoni effect, which pushes the liquid phase to flow away from low surface tension regions and was first explained by Gibbs [91]. A comprehensive experimental study on the flow maps by means of high-speed X-ray photography and flash methods simultaneously was reported by Hewitt and Roberts [66], which recognized the previous correlations for each flow pattern proposed by Baker [84] and Wallis [92]. A rough sketch of streamlines in the slug region of a GL flow was experimentally obtained by Taylor [31], where a stagnation point on the vortex and a stagnation ring on the curvature of the bubble were observed for two extreme cases of velocity fraction, (Vg − Vm)/Vg, greater and less than 0.5. The high-speed photography was compared with X-ray technology to provide clear images of air–water flow regimes through a capillary [66]. Irandoust and Andersson [34] experimentally depicted Taylor’s bubble profile, where the film thickness remained constant in the middle of the gas bubble. They also confirmed that the gas bubble was elongated in the front and squeezed in the rear parts. A review study on the flow maps was conducted by Akbar et al. [70] to show the dominant effects in each flow regime using available experimental data. They proposed a hydraulic diameter of 1 mm as a criterion for classifying the previous studies into the five regions: bubbly, plug or slug (surface tension dominated), annular (inertia dominated), dispersed, and transition zones regarding the amounts of superficial velocities and the Weber numbers. Micro-particle image velocimetry (µ-PIV) and fluorescence microscopy techniques were used by Günther et al. [73] to recognize a micro-segmented GL flow in a rectangular cross-section area channel. They showed that the roughness of the inner side of the channel and compressibility of the gas bubbles induce an imbalance into the flow field and accelerate the mixing rate accordingly. The periodic switching of recirculation regions in the liquid slugs caused a higher mixing rate in a meandering channel compared with a straight channel (Figure3). Dore et al. [ 93] investigated the dynamics of water/ionic two-phase flow by exploring slug formation and recirculation flow in fluid segments at T-junction, straight and curved microchannel employing µ-PIV. As shown in Figure4, the circulation patterns in water plug consisted of two main mirrored and counter rotating Processes 2021, 9, x FOR PEER REVIEWvortices for low mixture velocity. As the mixture velocity increased,12 of 45 two secondary vortices arose at the nose cap of water plug.

Figure 3. The 3. sketchThe ofsketch fluorescent of micrographs fluorescent for, (a) micrographs straight channel, and for, (b) (meanderinga) straight chan- channel, and (b) meandering nel [73]. Reproduced with permission from Günther, A.; Khan, S.A.; Thalmann, M.; Trachsel, F.; Jensen,channel K.F.,[ Lab73]. on Reproduced a Chip; published with by Royal permission Society of Chemistry, from Günther, 2001. A.; Khan, S.A.; Thalmann, M.; Trachsel, F.; Jensen, K.F., Lab on a Chip; published by Royal Society of Chemistry, 2001.

Processes 2021, 9, x FOR PEER REVIEW 13 of 45 Processes 2021, 9, 870 10 of 40

FigureFigure 4. 4.Schematic Schematic representative representative of of circulation circulation patterns patterns within within water water plugs plugs for for three three different different water water holdups holdups and and mixture mix- ture velocities [93]. Reproduced with permission from V Dore, D Tsaoulidis, P Angeli., Chemical engineering science; velocities [93]. Reproduced with permission from V Dore, D Tsaoulidis, P Angeli., Chemical engineering science; published published by Elsevier, 2012. by Elsevier, 2012.

AThe combination flow patterns of ofexperimental an LL two-phase observations flow in aand rectangular numerical cross-section simulations duct was were em- ployedexperimentally by Meyer investigated et al. [94] to byshow Zhao velocity et al. [profile,82]. The flow flow patterns maps and in a the 2 mm flow × transitions2 mm ver- ticalwere channel, correlated Figure by the 5. inertiaBoth methods forces of predicted each phase a Couette and the velocity interfacial distribution forces. They within realized the liquidsix distinctive film region, flow a mirrored maps in main terms recirculatin of dispersedg region, phase and droplet two small formation vortices process in the nose at a andT-junction rear caps as follows:with a direction of opposite. They also realized some differences between the results of the two methods as shown previously by [95]. • ExperimentalSlug regime; whenanalysis the by interfacial Zhao et al. tension [82] revealed is greater that than the inertial volume forces, of dispersed and the flow We- −6 −2 −6 was significantlyber numbers affected are 7.61 by× interfacial10 < We forces,ws < 4.87 inertia× 10force,and and5.94 the ×volumetric10 < Weratioks

Processes 2021, 9, x FOR PEER REVIEW 14 of 45

where the dynamic hold-up fraction (ε) and equivalent radius (R) in terms of the volume of dispersed phase (Vd) are,

ε = Qk/(Qw + Qk) (7) 3 −3 R = 3∀d/4π × 10 (8) The diameter of the channel has a major role in the measured void fraction and a flow pattern which is shown in Figure 6 for the small diameter tube of 38.1 mm (upper row) Processes 2021, 9, 870 and large diameter tube of 101.6 mm (lower row). An increase in tube diameter resulted11 of 40 in more compressed gas bubbles toward the top wall and occupied a smaller region of the cross-sectional area [96]. The measurement also showed that a further increase in superfi- cial velocityinertial ratio effects makes of the a more continuous uniform phase gas phase are significantly distribution greater realized than by a the smaller interfacial por- tion oftension. gas volume in the top region of the tube. As a consequence, the bubbly to plug/slug transition• Chaotic occurs thin striationsat lower superficial regime; when velocity both flow as the rates channel’s are increased diameter the We increases numbers (i.e., are 𝑗 = 0.51 m/s for the smaller pipe and 𝑗 = 0.25 m/s for the larger pipe). in a range of 0.17 < Wews < 30.43 and 4.29 < Weks < 53.5. This flow map will be Aeventually critical Reynolds changed number into annular of ⁓300 due was to thefound instability by Butler of theet al. flow. [97], where the time- averaged gas phase volume fraction showed a significant difference by means of the PIV A combination of experimental observations and numerical simulations was employed measurements in capillaries with a T-junction. As shown in Figure 7, a large mirrored by Meyer et al. [94] to show velocity profile, flow patterns in a 2 mm × 2 mm vertical recirculation region was observed between two consecutive O2 bubbles with a relative channel, Figure5. Both methods predicted a Couette velocity distribution within the liquid velocity of 2V – V resulting in the most efficient convective transport (regimes 1, 2, 11, film region, atp mirroredg main recirculating region, and two small vortices in the nose and andrear 13 caps in Figure with a 7). direction They also of revealed opposite. the They key alsorole of realized the recirculating some differences motion betweenin the liquid the plugresults on ofthe the dynamics two methods of mass as showntransfer previously and its dependency by [95]. on the bubble velocity.

FigureFigure 5. Flow patterns andand streamlinesstreamlines within within the the Taylor Taylor bubble bubble and and liquid liquid plug plug predicted predicted by ( aby,b) (µa-PIV,,b) µ-PIV, and ( cand) numerical (c) nu- mericalsimulation simulation [94]. Reproduced [94]. Reproduced with permissionwith permission from from Meyer, Meyer, C.; Hoffmann, C.; Hoffmann, M.; M.; Schlüter, Schlüt M.,er, M., International International Journal Journal of ofMultiphase Multiphase Flow; Flow; published published by by Elsevier, Elsevier, 2014. 2014.

Experimental analysis by Zhao et al. [82] revealed that the volume of dispersed flow was significantly affected by interfacial forces, inertia force, and the volumetric ratio of dispersed phase, which correlated by the multivariable least squares method, " # R We (1 − ε) = − ws + 0.1276 ln 0.15 0.5595 (6) dH (Weksε)

where the dynamic hold-up fraction (ε) and equivalent radius (R) in terms of the volume of dispersed phase (Vd) are, ε = Qk/(Qw+Qk) (7) p3 −3 R = 3∀d/4π × 10 (8) The diameter of the channel has a major role in the measured void fraction and a flow pattern which is shown in Figure6 for the small diameter tube of 38.1 mm (upper row) and large diameter tube of 101.6 mm (lower row). An increase in tube diameter resulted in more compressed gas bubbles toward the top wall and occupied a smaller region of the cross-sectional area [96]. The measurement also showed that a further increase in superficial velocity ratio makes a more uniform gas phase distribution realized by a smaller portion of gas volume in the top region of the tube. As a consequence, the bubbly to plug/slug transition occurs at lower superficial velocity as the channel’s diameter increases (i.e., jg= 0.51 m/s for the smaller pipe and jg= 0.25 m/s for the larger pipe). Processes 2021, 9, x FOR PEER REVIEW 14 of 45

where the dynamic hold-up fraction (ε) and equivalent radius (R) in terms of the volume of dispersed phase (Vd) are,

ε = Qk/(Qw + Qk) (7) 3 −3 R = 3∀d/4π × 10 (8) The diameter of the channel has a major role in the measured void fraction and a flow pattern which is shown in Figure 6 for the small diameter tube of 38.1 mm (upper row) and large diameter tube of 101.6 mm (lower row). An increase in tube diameter resulted in more compressed gas bubbles toward the top wall and occupied a smaller region of the cross-sectional area [96]. The measurement also showed that a further increase in superfi- cial velocity ratio makes a more uniform gas phase distribution realized by a smaller por- tion of gas volume in the top region of the tube. As a consequence, the bubbly to plug/slug transition occurs at lower superficial velocity as the channel’s diameter increases (i.e., 𝑗 = 0.51 m/s for the smaller pipe and 𝑗 = 0.25 m/s for the larger pipe). A critical Reynolds number of ⁓300 was found by Butler et al. [97], where the time- averaged gas phase volume fraction showed a significant difference by means of the PIV measurements in capillaries with a T-junction. As shown in Figure 7, a large mirrored recirculation region was observed between two consecutive O2 bubbles with a relative velocity of 2Vtp– Vg resulting in the most efficient convective transport (regimes 1, 2, 11, and 13 in Figure 7). They also revealed the key role of the recirculating motion in the liquid plug on the dynamics of mass transfer and its dependency on the bubble velocity.

ProcessesFigure2021 ,5.9, Flow 870 patterns and streamlines within the Taylor bubble and liquid plug predicted by (a,b) µ-PIV, and (c) nu-12 of 40 merical simulation [94]. Reproduced with permission from Meyer, C.; Hoffmann, M.; Schlüter, M., International Journal of Multiphase Flow; published by Elsevier, 2014.

Figure 6. Snapshot images to highlight the effects of channel’s diameter increase on bubbly to plug/slug transition regimes for different superficial velocities [96]. Reproduced with permission from Kong, R.; Kim, S.; Bajorek, S.; Tien, K.; Hoxie, C., Experimental Thermal and Fluid Science; published by Elsevier, 2018.

A critical Reynolds number of ~300 was found by Butler et al. [97], where the time- Processes 2021, 9, x FOR PEER REVIEW 15 of 45 averaged gas phase volume fraction showed a significant difference by means of the PIV measurements in capillaries with a T-junction. As shown in Figure7, a large mirrored recirculation region was observed between two consecutive O2 bubbles with a relative velocity of 2V –V resulting in the most efficient convective transport (regimes 1, 2, 11, Figure 6. Snapshot images to highlight the effecttp sg of channel’s diameter increase on bubbly to plug/slug transition regimes and 13 in Figure7). They also revealed the key role of the recirculating motion in the liquid for different superficial velocities [96]. Reproduced with permission from Kong, R.; Kim, S.; Bajorek, S.; Tien, K.; Hoxie, C., Experimental Thermal andplug Fluid on Sc theience; dynamics published of massby Elsevier, transfer 2018. and its dependency on the bubble velocity.

Figure 7. Time-averaged O2 volume fraction plots for 14 different flow regimes in terms of Re in the liquid plug regions Figure 7. Time-averaged O2 volume fraction plots for 14 different flow regimes in terms of Rebb in the liquid plug regions between two consecutive gasgas bubblesbubbles [97[97].]. ReproducedReproduced with with permission permission from from Butler, Butler, C.; C.; Lalanne, Lalanne, B.; B.; Sandmann, Sandmann, K.; K.; Cid, Cid, E.; E.; Billet, A.M., International Journal of Multiphase Flow; published by Elsevier, 2018. Billet, A.M., International Journal of Multiphase Flow; published by Elsevier, 2018.

TwoTwo different structures named high-conce high-concentratedntrated and and low-concentrated low-concentrated were were ob- ob- servedserved by YaoYao et et al. al. [98 [98]] to to characterize characterize the the mass mass transfer transfer and and flow flow patterns patterns of liquid–liquid of liquid– liquidslug flow slug at flow the bendat the region bend ofregion a rectangular of a rectangular meandering meandering microchannel. microchannel. As is shown As is shownin Figure in 8Figurea, smaller 8a, smaller flow rates flow created rates create a non-crossingd a non-crossing evolution evolution pattern pattern where where the red the redfilaments filaments occupied occupied the outerthe outer and innerand inner layers layers when when moving moving through through the bend. the bend. Conversely, Con- versely,when the when flow the rates flow were rates increased, were increased, Figure8 Figureb, the outer8b, the red outer filament red filament was shifted was shifted to the tocentral the central region region of the of bend. the bend. Table 5 presents selected experimental studies on the flow patterns of two-phase flows in tubular and non-tubular microchannels and gas bubble or droplet formation.

ProcessesProcesses2021 2021, 9, ,9 870, x FOR PEER REVIEW 1316 of of 40 45

FigureFigure 8. The 8. evolution The evolution patterns patterns of a typical of a typical water/toluene water/toluene two-phase two-phase flow at theflow bend at the of microchan-bend of mi- crochannel for (a) non-crossing pattern and (b) crossing pattern [98]. Reproduced with permission nel for (a) non-crossing pattern and (b) crossing pattern [98]. Reproduced with permission from from Yao, C.; Ma, H.; Zhao, Q.; Liu, Y.; Zhao, Y.; Chen, G., Chemical Engineering Science; published Yao, C.; Ma, H.; Zhao, Q.; Liu, Y.; Zhao, Y.; Chen, G., Chemical Engineering Science; published by by Elsevier, 2020. Elsevier, 2020. Table 5. The literature of selected experimental studies on the flow pattern maps and bubble or droplet formation. Table5 presents selected experimental studies on the flow patterns of two-phase flows in tubularComment(s) and non-tubular microchannels and gasCross-Section bubble or droplet formation.Phases Reference Several correlations as the function of velocity and pressure drop in each phase Circular GL [84] were suggestedTable 5. forThe different literature flow of selected regimes experimental studies on the flow pattern maps and bubble or droplet formation. Combination of X-ray and high speed flash photography technique Comment(s) Cross-SectionCircular PhasesGL Reference[66] High liquid flow rates Several correlations as the function of velocity and pressure drop in each phase Recognizing a novel flow map using Su number for microgravity two-phase flow Circular Circular GL GL [84 [65]] were suggested for different flow regimes Boiling heat transfer of R141b Combination of X-ray and high speed flash photography technique GL Heat transfer coefficient correlations Circular GL [66] High liquid flow rates Circular (single [99] Different flow regime in small- and large-diameter tubes phase) ObservationRecognizing of a novellocal dry-out flow map on using the channel Su number wall for microgravity two-phase flow Circular GL [65] UsingBoiling particle heat transfer image ofvelocimetry R141b GL RecirculationHeat transfer flow coefficient pattern correlations with a high degree of mixing Circular Circular (singleGL [99[37]] Different flow regime in small- and large-diameter tubes Counter-rotating vortices were observed inside the liquid slugs Square phase) VelocityObservation profile of localinside dry-out the slugs on the channel wall IncreasingUsing particle gas imagesuperficial velocimetry velocity led to the development of the slug flow LowRecirculation liquid superficial flow pattern velocity with made a high longer degree bubbles, of mixing shorter liquid slugs profile CircularCircular GLGL [37[68]] SlugCounter-rotating and annular vorticespatterns were observed inside the liquid slugs Semi-TriangularSquare AtVelocity high liquid profile superficial inside the velocity, slugs churn flow was established FiveIncreasing flow regimes gas superficial were observed velocity includ led toing the bubbly, development wedging, of the slug, slug bubbly, flow and Low liquid superficial velocity made longer bubbles, shorter liquid slugs profile Circular dry flow for moderate void fraction GL [68] TheSlug classification and annular of patterns patterns regarding liquid fraction Semi-TriangularSquare GL [72] At high liquid superficial velocity, churn flow was established The effect of a sharp return of the channel Void fraction measurement

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Table 5. Cont.

Comment(s) Cross-Section Phases Reference Five flow regimes were observed including bubbly, wedging, slug, bubbly, and dry flow for moderate void fraction The classification of patterns regarding liquid fraction The effect of a sharp return of the channel Square GL [72] Void fraction measurement The gravitational effects were taken into account for a bubbly pattern with a spherical gas bubble Liquid droplets may be observable on the walls in wedging pattern Using µ-PIV and fluorescent microscopy imaging The liquid phase segments were attached at the corners of the cross-section Rectangular GL [73] Gas phase flow improved the mixing and the residence time features The hemispherical ends of gas bubbles were not maintained as the Ca increases The nose and the tail of the bubble elongated and flattened as the Re increased Circular GL [44] The Marangoni effect was observed in experiment efforts and was not taken into account in the numerical simulations Surface modifications Contact angle measurements Meandering microchannel When the length of the bubble was equal to the channel diameter, the flow map Square GL [100] was annular At low void fraction when the surface energy was low, the isolated symmetric bubble patterns were observed At moderate void fractions, the flow pattern was asymmetric Discussed in the text Rectangular LL [82] An increase in Ca changes the droplet profile Development of droplet formation with the increase in Ca Circular LL [45] The significant deviation between the measured film thickness and the Bretherton and Taylor predictions At relatively low flow rates, the slug pattern was established with the length equal to the inner diameter of the tube Circular LL [101,102] At high volumetric flow rate, the deformed interface regime was made with long water slugs and small cyclohexane droplets Using a µ-PIV technique to capture bubble formation in a T-junction About 25% of the liquid phase passed the gas bubble The gas bubble formation process was included in three steps: Gas bubble growing until occupied the junction Square GL [103] Gas bubble developing was decreased as the liquid phase passes the bubbles and the bubble neck was tightened Gas bubble neck was rapidly decreased until approached one-quarter of the diameter before breaking up An empirical correlation for a transition from Taylor to unstable slug flow regimes At lower superficial velocities, the Taylor regime was established Circular GL [74,76] At higher superficial velocities, the Taylor flow was transited into the dripping flow pattern Boiling heat transfer of Fluorinert FC-77 As the heat flux enhanced, the bubble generation rate increased along with an GL increase of bubble length Rectangular (single [104] At moderate heat flux, flow pattern went back and forth between churn and phase) wispy-annular flow regimes Processes 2021, 9, 870 15 of 40

Table 5. Cont.

Comment(s) Cross-Section Phases Reference Bubbly flow regime was found at a low gas superficial velocity and a high liquid superficial velocity An increase in gas superficial velocity generated slug regime, where the gas-bubble length was longer than the diameter Circular GL [105] High liquid velocity and moderated gas superficial velocity produced churn flow map Low liquid superficial velocity and high gas superficial velocity transited the churn to the slug-annular PIV measurements were taken into account Backflow around the liquid slugs were observed Circular LL [106] The liquid slugs moved faster than that of average flow due to the lubricating effects Using a µ-PIV technique to capture flow structure Slug or Taylor pattern was observed for low superficial velocities As the superficial velocities increased, the lengths of bubbles, and slugs became Circular GL [107] more variable At high enough gas velocity, the gas phase penetrates the liquid and the length of slugs became shorter Using a laser signal system to provide flow pattern images The channel diameter affects the flow pattern and transition conditions Circular GL [108] In the bubbly flow regime, the size of the bubbles was approximately was the same as channel diameter A model of mass and momentum balance Stratified flow regime was observed for low oil to gas superficial velocity ratios Circular GL [109] An increase in the velocity of the gas phase makes the interface to be wavy The number of small bubbles in liquid plug/slug was significantly increased with increasing Vg The size of small bubbles was decreased with increasing Vg or increasing Vl Increasing Vg or decreasing Vl slightly increased the depth of the plug/slug bubble Circular GL [96] Increasing pipe size enhanced the contribution from large bubbles to a total void fraction Large bubble was accelerated due to small bubble coalescence as flow developed, leading to α decreases Indicating of wave growth and coalescence during slug flow formation For low superficial velocities, the flow pattern remained stratified smooth Circular GL [79] By an increase of superficial gas velocity, the flow pattern was changed to wavy

4.2. Numerical Simulations Numerical modeling methods can be categorized into four main groups; electronic, atomistic, mesoscale, and continuum [110]. In the following, we only review some of the widely-used numerical methods in the modeling of Taylor flows, and more in-depth reviews on the rest of numerical studies can be found in literature, i.e., [111]. Two different methods may be considered in numerical analysis when describing the motions of fluids in two-phase flows: continuous fluid (Eulerian) and discrete particle (Lagrangian). The first method solves the governing partial differential equations to predict the motion of the continuum-based fluid flow, while the second method follows the move- ment of fluid particles or molecules to predict fluid flow and heat transfer. The interaction between different phases is calculated by both methods to determine the coalescence and interference of interfacial forces [112]. Hashim et al. [113] simulated transport phenomena and biological components, such as proteins, DNA, and cells using the Eulerian method. The innovative aspect of their simulation was to describe the mixing concentration distribu- tion at the interaction of biological components. Apte et al. [114] numerically investigated Processes 2021, 9, 870 16 of 40

the liquid fuel spray from a nozzle using the Eulerian-Lagrangian approach with a point- particle approximation for the liquid droplets. Ni et al. [51] investigated numerical study of the GL and LL Taylor flows on an Arbitrary Lagrangian–Eulerian (ALE) formulation to track interfacial and curvatures of slugs. They showed the predominant roles of the viscosity ratio and density ratio of continuous phase to the disperse phase for quite large Ca causing a non-stagnant liquid film region. The Lagrangian approach was also used by [115–119], but not limited to. Therefore, these methods can be widely employed in applications concerning fluid motion. The Lattice Boltzmann Method (LBM) is a Computational Fluid Dynamics (CFD) method that simulates a fluid density on a lattice framework with streaming and collision (relaxation) processes instead of solving the Navier-Stokes equation directly [120]. This method is an efficient tool for modeling complex fluid systems that include complex bound- aries, microscopic interactions, propagations, and the collusions of particles [121–128]. The LBM in a GL two-phase flow modeling was proposed by Seta and Kono [129], based on a particle velocity-dependent forcing scheme. Their results revealed that the effects of high potential caused instability in a three-dimensional model. A diffuse interface free energy LBM was utilized by Komrakova et al. [130] to study the behavior of a single liquid-droplet suspended in another liquid. They found that for droplets with a radius of less than 30 lat- tice units, a smaller interface thickness is required. Li et al. [131] comprehensively reviewed different uses of LBM proposed by others for simulating thermal boundary treatments, interaction forces between two different phases, mechanical stability conditions, liquid- vapor phase changes, fuel cells, droplet collisions, and energy storage systems. Recently, Fei et al. [132] developed a three-dimensional, multiple-relaxation-time LBM based on a set of non-orthogonal vectors for modeling realistic multiphase flows. Some other uses of the LBM have also been developed by many investigators, e.g., [133–135]. Another widely used method for modeling two-phase flows is the Volume of Fluid (VoF), which is a numerical technique that determines the location of a free surface, GL, and LL interface by following the Eulerian approach. This numerical method is a powerful tool for modeling the interfaces of incompressible and immiscible two-phase flows because it is able to predict the interface between two phases even though the interface is becoming too weak to capture the curvature of the interfacial line [136]. Ketabdari [137] discussed VoF regarding the governing equations, as well as its advantages and capabilities predicting the interface line between two phases with large deformation. Osher and Sethian [138] devised a new numerical algorithm to realize and determine the propagation of curvature- dependent speed, which can be combined with VoF. This algorithm approximates the equations of motion by using hyperbolic conversation laws. Many investigators employed this method when simulating a two-phase flow’s interface, e.g., [139–142]. The general applications of this method and a comprehensive review were conducted by Osher and Fedkiw [143], which discusses the variants and extension methods, such as fast methods for steady-state problems, diffusion generated motion and the variation level set approach. The growth of liquid film thickness was governed by one equation [144]. They assumed a constant pressure gradient in the channel direction and a linear velocity distribution over the liquid film thickness, emphasizing that the model was valid only for a film thickness less than near-wall grid size. An effort to predict the bubble profile was analytically conducted by Bretherton [30], where he found that considering gravitation as a buoyancy force produced nonsensical   results in the vertical configuration of a capillary tube. A critical value ρgR2/σ of 0.842 was introduced to show no upward movement of the bubble for the amount less than the critical value. Kolb and Cerro [145] found an axisymmetric bubble profile for the Ca greater than 0.1, where the lubrication’s law predicts the interface profiles and the flow patterns adequately. The effects of bubble and slug lengths on the mass transfer in a GL flow was experimentally studied by Berˇciˇcand Pintar [146]. Their results revealed that the amount of nitrite transported was independent of the gas bubble length and primarily depended on the liquid slug length. The flow field predictions were numerically obtained Processes 2021, 9, 870 17 of 40

by Giavedoni and Saita [36], who illustrate the streamlines for showing the bubble profiles and the slug region. They found that with an increase in the Ca, the recirculation flow disconnected the GL interface and moved downstream. Brauner et al. [147] followed an exact analytical solution to model the interface curvature of two-dimensional stratified flow. Heil [43] numerically simulated the influence of fluid inertia, Re, on the flow patterns and the propagation of an air bubble on two flexible and rigid walls of a GL flow in a two- dimensional channel. Fujioka and Grotberg [148] numerically analyzed the propagation of liquid plugs inside a two-dimensional channel using a finite volume method. They showed that as the liquid plug grew, the frontal part swept the interfacial surfactant from the precursor liquid film. A pair of recirculation regions was found inside the liquid plug shortening as the magnitude of the velocity decreased. They also noted that micelle production and its transport process must be added to the numerical model when plug propagation occurs with a much higher concentration. Their results revealed that as the Re increases, a recirculation flow appears inside the plug core and is then skewed toward the rear part of the bubble. Kreutzer et al. [44] denoted that the inertia effects due to an increase in the Reynolds number elongated the nose and flattened the rear menisci of the gas bubble profile. The influence of pressure gradient on the flow patterns and bubble profile were illus- trated in Figure9. According to this figure, as the pressure gradient increases, the elongated gas bubble is stretched along the tube axis and the impact of the no-slip conditions at the walls becomes weaker on the bubble shape. Strong, and clockwise circulation is predicted next to the head cap of the bubble resulted in bubble elongation when the pressure is intermediate. Falconi et al. [149] numerically and experimentally studied a vertical-upward three-dimensional Taylor flow in a square mili-channel using the VoF in-house finite vol- ume code and µ-PIV measurements, respectively, Figure 10a. Instantaneous streamlines in Figure 10b show three vortices within the bubble; a large central vortex, two small toroidal vortices at the caps of the bubble. The main central vortex represents the highest z-component velocity next to the channel axis with the direction of rotation of opposite to the smaller vortices. Valizadeh et al. [150] conducted a numerical parametric study of non-Newtonian turbulent flow in a spiral duct by investigating the effects of geometry, consistency index, and power-law index values of viscosity. Since their study was dealt with a wide range of large Reynolds numbers in mili-channel, we do not discuss their findings anymore. Both experimental and 3D numerical simulations were carried out by Abdollahi et al. [151] for liquid–liquid Taylor flow in a square channel with hydraulic diameters of 1 and 2 mm. Figure 11 displays the non-dimensional contour of the radial velocity component highlighting its key effects on heat transfer enhancement and recirculating flow within the slugs and the carrying phase. The highest radial velocity occurred at the nose and rear of the slug to make two vortices rotating in the opposite direction of the main recirculation zone in the middle of the slugs as observed by [139,149]. In a constant length unit cell, as the dispersed phase volume fraction increased, the liquid film thickness remained constant. The large recirculating zone in the middle of the slug show less axial flow and more radial flow as the droplet length was decreased. Their results also showed a huge increase of heat transfer rate up to 700% compared to single-phase flow indicating more effectiveness of short slugs on the heat transfer rate. ProcessesProcesses2021 2021,,9 9,, 870 x FOR PEER REVIEW 1821 ofof 4045

Processes 2021, 9, x FOR PEER REVIEW 21 of 45

FigureFigure 9.9. The streamlines inside and outside outside the the bu bubblesbbles for for different different pressure pressure gradients gradients (a (a) )85 85 M M −1 −1 −1 PamPamFigure−1, ,((bb) ) 9.850 850 The M M streamlines Pam Pam−,1 and, and inside (c ()c 3000) and 3000 outsideM M Pam Pam the− [139].1 bu[139bbles Reproduced]. Reproducedfor different with pressure with permission permission gradients from ( froma) 85 Fukagata, Fukagata,M K.; Kasagi,Pam−1, (b N.;) 850 Ua-arayaporn, M Pam−1, and ( cP.;) 3000 Himeno, M Pam T.,−1 [139]. International Reproduced Journal with permissionof Heat and from Fluid Fukagata, Flow; pub- K.; Kasagi, N.; Ua-arayaporn, P.; Himeno, T., International Journal of Heat and Fluid Flow; published lishedK.; Kasagi,by Elsevier, N.; Ua-arayaporn, 2007. P.; Himeno, T., International Journal of Heat and Fluid Flow; pub- by Elsevier,lished by 2007.Elsevier, 2007.

FigureFigure 10. (a 10.) The (a) The schematic schematic of of computational computational setup setup and and coordinate coordinate system, system, and (b and) instantaneous (b) instantaneous streamlines streamlines in the mov- in the ing frame with the bubble and z-component velocity contour-plot in a fixed frame of reference [149]. Reproduced with moving frame with the bubble and z-component velocity contour-plot in a fixed frame of reference [149]. Reproduced withFigure permission 10. (a) The from schematic Falconi, of C.J.; computational Lehrenfeld, setup C.; Marschall, and coordinate H.; Meyer, system, C.; Abiev,and (b R.;) instantaneous Bothe, D.; Reusken, streamlines A.; Schlüter, in the mov- M.; Wörner,ing frame M., with Physics the bubble of Fluids, and published z-component by AIP velocity Publishing, contour-plot 2016. in a fixed frame of reference [149]. Reproduced with

Processes 2021, 9, x FOR PEER REVIEW 22 of 45 Processes 2021, 9, x FOR PEER REVIEW 22 of 45

Processes 2021, 9, 870 19 of 40 permission from Falconi, C.J.; Lehrenfeld, C.; Marschall, H.; Meyer, C.; Abiev, R.; Bothe, D.; Reusken, A.; Schlüter, M.; permission from Falconi, C.J.; Lehrenfeld, C.; Marschall, H.; Meyer, C.; Abiev, R.; Bothe, D.; Reusken, A.; Schlüter, M.; Wörner, M., Physics of Fluids, published by AIP Publishing, 2016. Wörner, M., Physics of Fluids, published by AIP Publishing, 2016.

Figure 11. Radial velocity component contours with different dispersed phase volume fraction in a fixed unit cell length Figure[151]. 11. 11.Reproduced RadialRadial velocityvelocity with component permissioncomponent contoursfrom contours Abdollahi, with with different different A.; Norri dispersed disperseds, S.E.; phaseSharma, phase volume R.N.,volume fraction Applie fractiond in Thermal a fixedin a fixed unit Engineering; cellunit length cell length pub- [151]. [151].lished Reproduced by Elsevier, with 2020. permission from Abdollahi, A.; Norris, S.E.; Sharma, R.N., Applied Thermal Engineering; pub- Reproduced with permission from Abdollahi, A.; Norris, S.E.; Sharma, R.N., Applied Thermal Engineering; published by lished by Elsevier, 2020. Elsevier, 2020. Xu et al. [152] numerically simulated 3D Taylor flow in a microchannel with a square cross-sectionXu et al. [152] [152under] numerically numerically ultrasonic oscillation. simulated 3DA harmonic, Taylor flowflow vertical, in a microchannel and ultrasonic with oscillating a square cross-sectionapplied to microreactor under ultrasonic as, oscillation. A harmonic, vertical, and ultrasonic oscillating applied to microreactor as, applied to microreactor as, dy = A cos (ω0t) (9) dyd= A= A cos cos( ω(ω0t0)t) (9) where the amplitude of the microreactory oscillation (A) is in the order of 2–8 microns and where the amplitude of the microreactor oscillation (A) is in the order of 2–8 microns and wherethe angular the amplitude frequency of as the a functionmicroreactor of a constantoscillation vibrating (A) is in frequency the order of of f2–80 = 20 microns kHz is and the angular frequency as a function of a constant vibrating frequencyfrequency ofof ff0= = 20 kHz is is 0 ω0 = 2πf0 (10) ωω= =2 π2πf f (10) They found the channel oscillation affected0 0 the00 flow pattern and hydrodynamic char-(10) acteristicsThey found foundin both the the liquid channel channel slugs oscillation and oscillation bubbles affected affected as shown the theflow in Figure flow pattern pattern 12 and for Taylorhydrodynamic and hydrodynamic flow in bends char- acteristicscharacteristicsand oscillating in both in bubbly bothliquid liquid flows, slugs slugs andfor example. bubbles and bubbles asTwisted shown as shownflow in Figure structure in Figure 12 for in 12Taylor the for liquid Taylor flow slugs in flow bends im- in andbendsproved oscillating and mixing oscillating bubbly rate and bubbly flows, transport flows,for example. forphenomena. example. Twisted TwistedWavy flow interface flowstructure structure of in bubble the in liquid the enhanced liquid slugs slugs theim- provedimprovedinterfacial mixing mixing area rateaccelerating rate and and transport transport the mass phenomena. phenomena. transfer rate Wavy Wavy across interface interface the channel of of bubble bubble and improved enhanced the the interfacialperformance area area of accelerating microreactor. the The mass sub-harmonic transfer rate bubble across surface the channel wave was and also improvedimproved created thetheby performancethe pressure ofpulsation microreactor. in the Theparallel sub-harmonic direction of bubble the oscillation. surface wave was also created by the pressure A summary pulsation of analytical in the parallel and numerical direction investigations ofof thethe oscillation.oscillation. on the flow patterns and the formationsA summary of the of gas/liquid analytical slugs and isnumerical presented investigations in Table 6. on the flow patterns and the formations of the gas/liquid slugs is presented in Table 6.

Figure 12. Schematic of different gas–liquid Taylor flow in microchannel; Taylor flow in bends,

oscillating bubbly flow, and oscillating Taylor flow observed by [152]. Reproduced with permission from Xu, F.; Yang, L.; Liu, Z.; Chen, G., Chemical Engineering Science; published by Elsevier, 2021. Processes 2021, 9, 870 20 of 40

A summary of analytical and numerical investigations on the flow patterns and the formations of the gas/liquid slugs is presented in Table6.

Table 6. Selected numerical and analytical studies on the flow patterns and the bubble or the droplet formation.

Comment(s) Cross-Section Phases Reference The film thickness at the advance of the rear meniscus oscillated The front and rear menisci were slightly different in curvatures Circular GL [30] Equilibrium bubble profile under surface tension and gravitational effects in a vertical tube The film thickness The profiles of front and rear film thickness Circular and Square GL [35] The film thickness for very elongated gas bubbles Bubble profile Flow fields around and between bubbles Square GL [145] The motion of a bubble The influence of velocity on the mass transport phenomenon Circular GL [146] The effects of liquid slug lengths on flow parameters Stronger recirculating flow region as the Ca decreased A liquid backflow was appeared at non-dimensional liquid thickness less than 1/3 Parallel plates and Single stagnation point at the vertex of bubble curvature for no-recirculating GL [36] Circular flow conditions A decrease in the Ca made recirculating flow and moved the stagnation point further the vertex The effects of flow rates on the interface curvature Two-phase Circular [147] The influence of Eo and wall adhesion on the interface curvature (parametric) The rear profile of bubble meniscus versus Ca and Re The effect of Re on the free surface undulations Circular GL [38] Gas bubble profile The curvature of the gas bubble The flow inertia effect was more significant even for deformable wall channels Flexible wall channels were more sensitive than rigid wall channels to the 2D GL [43] propagation of air bubbles into the walls at low Re and Re/Ca situations The pressure gradient in faraway positions of the tip of the air bubble was generated by Poiseuille flow The propagation of liquid plug The adsorption/desorption process of the surfactant was modeled 2D GL [148] The Marangoni stress results in nearly zero surface velocity at the front meniscus Refer to Table5 Circular GL [44] Slug flow development with time Either uniform or parabolic inlet velocity profile made the same-sized slugs 2D GL [58] The inlet mixing level influence on the slug lengths Bubble shape in slug flow regime The effects of pressure gradient on the bubble profile 2D GL [139] The presence of gas bubbles made the circulating regions stronger causing a higher momentum transport Curved vertical microchannel Slug flow development with time Circular GL [153] The gas bubbles moved faster than liquid slugs The impact of inlet geometry on the slug development Processes 2021, 9, 870 21 of 40

Table 6. Cont.

Comment(s) Cross-Section Phases Reference The VoF and level-set methods 2D LL [106] The lack of enough adhesion on wall deformed the rear interface of the slugs The VoF method Bubble shapes and formation 2D and 3D GL [154] The effects of superficial velocities on the bubble profiles The influence of nozzle diameter on the bubble formation and shapes Structured-square grid minimizes inaccuracies in the surface tension calculation 2D GL [155] The process of bubble formation was happened periodically The VoF and level set techniques Two commercial simulating software; Ansys and TransAT 2D GL [156] Bubble formation and development The effect of mixture velocity on the flow pattern and the bubble shapes Bubble curvature in slug flow The characteristics of flow were dominantly determined by adherent liquid film thickness 2D GL [26] An optimized model to minimize the pressure drop and maximize the heat transfer rate

5. Flow Regime Transitions A transition from one flow regime to another often occurs when the flow or boundary conditions are changed. The actual flow maps and the transition conditions specifically depend on a group of effective features during an experiment or the assumptions in numerical simulations [157–160]. For example, Bottin et al. [161] experimentally obtained various flow regimes for a horizontal two-phase flow in a pipe with an inner diameter of 0.1 m. They attempted to describe different flow regimes regarding the values of superficial velocities of the gas and liquid phases in the earlier experiments [162–167]. Some of the transition criteria from a flow regime to another in microchannels are presented in Table7. Jayawardena et al. [65] showed the ratio of gas to liquid Reynolds numbers as a function of the Suratman number for recognizing the flow maps. An accurate transitional value of the Su occurs between 104 and 107 from a bubble to a slug transition. However, the transition from a slug regime to an annular regime has two different criteria regarding the value of Su, greater and less than 106. The motion of the gas bubble through a microchannel with a square cross-sectional area was experimentally studied by Cubaud and Ho [72] and Cubaud et al. [100]. Regarding the liquid fraction, Ql / (Ql + Qg), they classified the flow patterns into several categories: bubbly, wedging, slug, annular, and dry, which are illustrated in Figure 13. Their experiments specified some critical liquid fractions for a transition from one flow regime to another, which are ~0.75 from bubbly to wedging, ~0.20 from wedging to slug, ~0.04 from slug to annular, and ~0.005 from annular to dry transitions. They also found that at low liquid velocity, the gas bubbles clogged at the sharp corner of the channel bend before merging and eventually passing the bend. The corner of the channel bend trapped a gas bubble to merge into another upcoming bubble, producing a single larger bubble. ProcessesProcesses2021 2021, 9, ,9 870, x FOR PEER REVIEW 2225 of of 40 45

FigureFigure 13. 13. The different flow flow patterns patterns from from high high to to low low liquid liquid fraction fraction [100] [100 (Reproduced] (Reproduced with with permissionpermission fromfrom Cubaud,Cubaud, T.; Ulmanella Ulmanella U.; U.; Ho, Ho, C.M., C.M., Fluid Fluid Dynamics Dynamics Research; Research; published published by by Elsevier,Elsevier, 2006). 2006).

Table 7. Summary of proposed criteria for the transition of flow pattern. Table 7. Summary of proposed criteria for the transition of flow pattern. Cross-Sec- Refer- Comment(s)Comment(s) CriterionCriterion Mechanism Mechanism Cross-Section PhasesPhases Reference tion ence Bubbly to slug transition The collision frequencyBubbly was to slug so low transition when When the rate ofWhen the rate of theThe void collision fraction frequency was smaller was than so 10% low whencoalescence the void wascoalescence much Bubblewas much coalescence Bubble coalescence The collisionfraction frequency was was smaller rapidly than 10%more than that ofmore break-up than that of break-and Circular GL [168] and Circular GL [168] increasedThe collision at a void frequency fraction above was 25%rapidly so increasedand at a at void a fraction of up void fraction that transition to slug flow was rapid even 0.25 − 0.3 void fraction void fractionin a above strongly 25% liquid so that transition to slug flow and at a void fraction of was rapid even in a strongly liquid 0.25 − 0.3 Minimum gas and V ρ0.5 Churn to annular happens at a value of g g 0.5liquid superficialMinimum gasCircular and GL [92] 0.5 Vgρ [gd(ρ − ρ )] g Churn to annular happens at a value of l g 0.5velocities liquid superficial ve- Circular GL [92] gd ρl − ρg Slug to slug-bubbly transition ReWe =2.8 × 105 locities No stratified flow pattern was observed in where, The production5 of Re Slug to slug-bubbly transition ReWe = 2.8 × 10 Rectangular GL [64] channels with diameter less than 1 mm Re = ρ dVg / 2µ and the We numbersThe production of No stratified flow pattern was observed in channelsl l where, (it was also verified by [67]) We = ρ dV2 / 2 l g Re = ρ dV / 2µ Re and the We num- Rectangular GL [64] The churn to annularwith diameter transition less occurred than 1 mm l g l 2 bers (itat was lower also verified by [67]) We = ρ dV / 2 the same as [92]l Theg same as [92] Circular GL [66] Thein comparison churn to annular to the flow transition reversal occurred at lower transition the same as [92] The same as [92] Circular GL [66] in Bubblycomparison to slug to transition the flow reversal transition 0.25 Void fraction Circular GL [163] Bubbly to slug transition 0.25 Void fraction Circular GL [163] π Kelvin-Helmholtz H ≥ 1 − d The transition from stratified to slug regimes 4 instability and criti- Circular GL [164] Bocr = 4.7 cal Bond number Upward and vertical flow For bubbly to slug: Void fraction Circular GL [169]

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Table 7. Cont.

Comment(s) Criterion Mechanism Cross-Section Phases Reference Kelvin-Helmholtz The transition from stratified to slug H ≥ 1 − π
d 4 instability and critical Circular GL [164] regimes Bo = 4.7 cr Bond number Upward and vertical flow For several transitions For bubbly to slug: The flowing quality was more suitable Void fraction Circular GL [169] ~0.3 than the thermal equilibrium quality to correlate the flow-regime transitions Concentric and eccentric annuli (eccentricity 50%) Upward and vertical flow V Vmax Probability Density Mathematical modeling and Circular GL [170] Function (PDF) Insignificant effect of eccentricity on R PDF dv transitions for many transitions

The dominant surface tension in stratified 2 Linear stability (2π) σ Circular GL [171] o = 2 flow regime (ρl − ρg)d g analysis Bubble to slug transition Upward and vertical flow ∼ 34% Average void fraction Circular GL [172] Turbulence-dependent rate processes controlled the transition at a high flow rate

Reg − 2 = 464.16 Su 3 Rel Discussed in the text ( Reg − 2 Suratman number Circular GL [65] = 4641.6 Su 3 Rel −9 2 Reg= 2 × 10 Su Bubble to slug transition (N/A) Void fraction Circular GL [173] Upward and vertical flow Bubbly to slug transition The formulation of interfacial transport equations Interfacial area (N/A) Circular GL [174] The classification of bubble interactions concentrations The categorization of basic mechanisms of bubble coalescence and breakup

for s < db α = 0.2 Upward and vertical flow for a bubble to for d ≤ s ≤ 3d b b Void fraction Rectangular GL [175] slug transition α = s +0.15 20 db for s > 3db α = 0.3 Discussed in the text Discussed in the text Liquid fraction Square GL [72,100] The bubble size distribution played a Population Balance crucial role in the flow regime transition (N/A) Circular GL [176] Model (PBM) and flow patterns Slug into parallel transition happened at small flow rates and large superficial flow ratio Specific range of We for By increasing the flow rate, the chaotic Weber number Rectangular LL [82] each flow pattern flow was changed into annular Equivalent radius was correlated to the We numbers and holdup fraction Slug to deformed interface occurred for 2.5 < Q /Q < 5 Volumetric flow rate c d Qc Circular LL [45] Slug to drop transition happened in the Qd ratio 3 < Qd /Qc < 6 Bubbly to slug transition Vertical and upward configuration Maximum bubble size PBM Circular LL [101] Effect of bubble size and diameter on the transition Processes 2021, 9, 870 24 of 40

Table 7. Cont.

Comment(s) Criterion Mechanism Cross-Section Phases Reference Bubbly to slug transition The influence of the bubble population at the inlet region on the regime transition An increase in bubble diameter transits Bubble diameter PBM Circular GL [177,178] the flow map from bubbly to slug A decrease in tube diameter changed the flow patterns from slug to bubbly regimes Stable slug regime in the square duct at Reg < 24 Stable slug regime in the circular duct at Reg < 36 Square Stable slug regime in the rectangular duct ρ Reynolds number of gVgdH and GL [77] at Reg < 40 µg gas phase Stable slug regime in the trapezoidal duct Trapezoidal at Reg < 72 Annular flow regime at Rel ≤ 24Reg − 7638 Cal Slug to churn transition and flow map α ≥ f Void fraction Circular GL [179] identification α A transition on the slug producing from Gas volumetric flow squeezing to dripping occurred at the Q > 20 Rectangular GL [180] g rate junction Above the transition value of Ca, an elliptic or oval cross-section area of the gas Quasi- 0.01 Capillary number GL [80] bubble was observed, and viscous shear trapezoidal overcame the interfacial tension

6. Slug Length The hydrodynamic characteristics of slug flows are highly dependent on the profile and length of the liquid slug. Bubble length measuring was experimentally conducted by Marchessault and Mason [29], where the change in the shape of the liquid film thickness around short bubbles was likely caused by end effects, buoyancy, and cylindrical model assumption. According to Schwartz et al. [32], the liquid film thickness left behind a gas bubble in a capillary was adequately close to the predictions of lubrication theory when the length of bubbles was short. They also found that for the length of bubbles shorter than the tube diameter, the thin film layer region was no longer observed. According to Kashid and Agar [101] and Kashid et al. [102], slug length is significantly affected by the capillary number and the junction dimensions. In a slug flow regime, the lengths of slugs of both phases are greater than their diameters, but at relatively low volumetric flow rates the slug pattern is established with the length equal to the inner diameter of the tube. In the case of a high volumetric flow rate and surrounded water slugs by cyclohexane, the deformed interface regime is made with long water slugs and small cyclohexane droplets. A few simple linear equations to compute the stable slug lengths were reported by [163,170,181,182], which indicated that the ratio of the liquid slug length to the tube diameter remained at a constant value of 16, 20.7, 7.5, and 21.5, respectively. Table8 presents the correlations for measuring the length of liquid slugs of two-phase flow in a microchannel including the void fractions. Beyond the data provided in Table8 , two other recent studies, Han and Chen [183] and Qian et al. [184], are concerned with the slug length and its formation, but the correlations of slug length were not proposed by the authors. The first study numerically simulated the droplet formation at a T-junction in a microchannel. The effective diameter of droplets was studied in terms of contact angle, continuous phase viscosity, flow rate ratio, and interfacial tension. The latter experimental investigation considered the influence of flow rate on the liquid slug generation in a microchannel with a square cross-sectional area. They found the slug lengths vary from 0.923 to 1.186 mm based on different flow rates of the continuous and dispersed phases. The distance between subsequent water slugs was also measured. Processes 2021, 9, 870 25 of 40

Table 8. The literature on the correlation of the slug length in two-phase flows in circular and non-circular cross-section area microchannels.

Correlation Comment(s) Flow Parameter Cross-Section Phases Method Reference

 0.63 The gas and liquid slug L  0.63 ρ σ2 g = Re = 0.0878 Vs l lengths versus d ≤ 4 mm d Eo2 µ ρ −ρ 2 3 2 l( l g) d g dimensionless numbers 55 ≤ Re ≤ 2000 Circular GL Experimental [185]  1.2688  1.2688 µ σ The effect of gas superficial 0.0015 ≤ Ca ≤ 0.1 Ll = 1 = l d Re Eo 3451 3 (ρl−ρg)ρlVgd g velocity on slug lengths The slug length was extensively depended on d = 50 mm 1−ε d3 physical means of the −1 −1 L = g π g 0.07ms ≤ Vg ≤ 0.27 ms Circular GL Experimental [186] l εg 6 2 experiment, operational d 0.07 ms−1 ≤ V ≤ 0.27 ms−1 conditions, and delivering l system w = 0.1 mm The squeezing mechanism h = 0.033 mm L Qd LL = 1 + α affected the size of droplets 10 mPas ≤ µc ≤ 100 mPas Rectangular Experimental [187] w Qc GL or bubbles 10 mPas ≤ µd ≤ 100 mPas −1 Qc= 0.00278, 0.0278, 0.278 µLs A decrease in superficial liquid velocity increased the gas slug length An increase in superficial gas velocity enhanced the Luc = ε−0.893 − ε −1.05 −0.075 −0.0687 d = 0.25, 0.5, 0.75, 1, 2, 3 mm d 1.637 (1 ) Re Ca gas slug length 2D Lg −1.05 − − 15 ≤ Re ≤ 1500 = 1.637ε0.107(1 − ε) Re 0.075Ca 0.0687 Both slug lengths were − and GL Numerical [58] d 2.78 × 10 4 ≤ Ca ≤ 0.01 Ll −0.893 −0.05 −0.075 −0.0687 moderately affected by the Circular = 1.637ε (1 − ε) Re Ca 0.09 ≤ ε ≤ 0.91 d surface tension and wall surface adhesion The gravitational effects on the slug lengths can be ignored Processes 2021, 9, 870 26 of 40

Table 8. Cont.

Correlation Comment(s) Flow Parameter Cross-Section Phases Method Reference The effects of wettability and the shape of the interface as the volumetric flow rates w = 0.4 mm  −0.2 ratio on the plug length Ld = Qc −0.2 Ca < 0.1 Square GL Experimental [59] w 1.59 Q Ca The equilibrium of the shear d Qc > 1 force and interfacial tension Qd in the form of Ca on the plug length 0.2 mm ≤ w ≤ 0.4 mm An increase in superficial h = 0.15 mm velocities enhanced the slug × −3 ≤ ≤ × −3 " V # 4.7 10 Ca 7.4 10 Experimental  0.33 l length L Re Vl+Vg 11.6 ≤ Re ≤ 22.6 Rectangular GL and [60] l = 0.369 ln +3.15 Larger bend diameter of the w Ca 509.12 20.4 × 103 ≤ Pe ≤ 39.6 × 103 Numerical meandering channel made −1 −1 0.01 ms ≤ Vl ≤ 0.07 ms longer slugs −1 Vg = 0.08 ms The inlet conditions affect d = 0.5 mm the bubble length − V = 0.245 ms 1 Lg = g d 2 significantly −1 Vl = 0.255 ms 2D GL Numerical [155] Ls = The lengths were computed d 1.3 Re = 280 at the centerline of the Ca = 0.006 channel Droplet size at the junction  w 0.5 d3D h −0.89 −0.5 = 0.334 w Cac The dependency of droplet w = 22.6 and 23.7 µm h h +0.79 w > diameter on the channel h = 5 µm Rectangular LL Experimental [61] and at h 4.5: width was insignificant at 0.008 ≤ Ca ≤ 0.2 d3D = 0.334 Ca−0.5 c h c large width to height ratio

dH= 0.15, 0.29, and 0.4 mm Variation of slug length with 0.1 ≤ We ≤ 26 L − −1 −1 g = ε0.07 − ε 1.01 −0.1 volumetric gas and liquid 0.02 ms ≤ Vg ≤ 1.2 ms Square GL Experimental [188] d 1.3 (1 ) We H −1 −1 flow rates 0.004 ms ≤ Vl ≤ 0.7 ms 0.06 ≤ ε ≤ 0.85 Processes 2021, 9, 870 27 of 40

Table 8. Cont.

Correlation Comment(s) Flow Parameter Cross-Section Phases Method Reference

L V   w = 0.75 mm g = g 1+1.37 We−0.349 w Vl The effects of flow rates on h = 0.28 mm  −0.365 −0.373 −1 −1 Rectangular GL Experimental [189] Ll Vg Vl −0.208 the slug lengths 0.08 ms ≤ V l ≤ 0.5 ms w = 1.157 V +V V +V We −1 −1 l g l g 0.155 ms ≤ Vg ≤ 0.952 ms

s The slug lengths of  −   Q  1.5 = 1± 1−4w 0.96 d dispersed and continuous w 0.1 mm Qc h = 0.095 mm L =   phases d  Q −1.5 −1 −1 2 0.96 d The slug formation process 4 µLmin ≤ V ≤ 44 µLmin Rectangular LL Experimental [190] Qc l Qd  Q −1.1 was not affected by pressure 0.6 ≤ ≤ 3 0.73w d Qc Qc fluctuations µ = 1 cs Lc = L d d Varying slug lengths The length of dispersed slugs The effects of operational d = 0.6, 0.8, 1 mm Q  Q 0.89 conditions on the slug ≤ a ≤ = −0.32 1.25 a 0.5 Q 3 Circular LL Experimental [191] Ld 0.0116Vuc d Q lengths o o −1 ≤ ≤ −1 The influence of slug 0.01 ms Vl 0.03 ms velocity and flow rate on the slug length w = 0.8 mm A slug flow in h = 0.9 mm superhydrophobic 6 < Re < 40 = 1 ( + ) microchannel − Rectangular GL Experimental [180] Ll 2 tCH1 tCH2 Vl 5 mLmin 1 ≤ Q ≤ An increase in gas flow rate g 30 mLmin−1 decreased the slug length Q 2 ≤ g ≤ 300 Ql Processes 2021, 9, 870 28 of 40

7. Pressure Drop The interactive structure of different phases and between phases and solid walls represent the predominant parameters to describe the flow pattern of multiphase flow and pressure loss through the channel. Two homogenous and separated flow models have been employed by researchers to predict the frictional pressure drop [62]. The first model postulates the same velocity for both gas and liquid phases, which implies that the slip ratio at the interactive boundaries is equal to one. This model considers two or more different phases as a single phase. The values of the flow properties are dependent on the quality, while the frictional pressure drop can be obtained by single-phase theory [63]:

 1 d   dp  = f 2 (11) 2 ρV dz l where z indicates the axial direction of the channel. The Hagen–Poiseuille equation is a physical law to compute the pressure drop of a Newtonian and incompressible flow through a long and circular tube with a constant cross-section area [63]. The flow regime remained laminar due to the Reynolds number of the microflows, where the friction factor becomes f = 16/Re for round tubes and the equation above can be rearranged as:

16  1  4L ∆p = ρV2 (12) Re 2 d

where L and d are the length and the diameter of the tube, respectively. The presence of the second phase in two-phase flow causes an additional term to the pressure loss of a single-phase flow and does not satisfy Poiseuille’s law anymore.
Therefore, the total pressure drop (∆pt) is the sum of the single-phase pressure drop ∆ps and the interaction pressure drop (∆pint) caused by introducing the secondary phase. These pressure drop components can be shown together as follows:

∆p∗ fRe = 16+ int (13) t 2L∗ The non-dimensional pressure drop and length scale in Equation (12) were proposed by Walsh et al. [54], as below: ∆p d ∆p∗ = int (14) int µV L L∗ = s (15) d

where µ and Ls denote the dynamic viscosity of the continuous phase and the length of liquid slugs, respectively. The pressure drop over a unit cell can also be described by three components [51,88,107]:

∆puc = ∆ps + ∆pf + ∆pcap (16)

Furthermore, Abiev [192] assumed the total pressure losses in the capillary tubes were a summation of two components, named the formation of a new surface during the motion of bubble and the arrangement of a velocity profile in liquid slugs as below (refer to Table9 for further details): ∆pt = ∆p∆F + ∆ptrans (17) Processes 2021, 9, 870 29 of 40

Table 9. The literature on pressure drop correlations of the two-phase flow in the microchannels.

Correlation Comment(s) Flow Parameter Cross-Section Phases Method Reference Dynamic pressure component was dominantly greater than the static value (lubrication d = 1 mm Experimental Across a bubble approximation) for −3 σ 2 1 × 10 ≤ Ca ≤ Circular GL and [30] = 3 small Ca ∆p 14.894 d Ca 0.01 The main pressure drop Analytical caused by a slight difference between the head and rear curvatures of the bubble The end effect (K) of the bubble was shown by 0.5 mm ≤ d ≤ a constant value of 45 2µ V h i 0.8 mm ∆p = l g 64ν +K for 0 < Re < 270 ≤ ≤ Circular GL Experimental [64] d πd3 0.5 Us/Ub 1 a function of the Re of 23 ≤ µl 0.163(ρ lVg D)/2µl for µb ≤ 58 270 < Re < 360 For circular tube: 2 ∆p =9.4 Ca 3 − 12.6 Ca0.95 Matched Asymptotic For Square tube: 3 × 10−3 ≤ Ca ≤ Numerical axisymmetric bubbles, Analysis Circular and GL and [35] > 0.01 Square = 0.14 Infinite bubbles, L d Bo = 4 × 10−4 Analytical ∆p 3.14 Ca Finite bubbles, L < d non-axisymmetric bubbles, ∆p =12.2 Ca0.55 Numerical   −0.66 An entry region friction Ls (N/A) Circular GL and [39] ∆p =∆p εl 1 + 0.065 model l d Rel Analytical Bubble profiles For 0.1 < Ca < 0.54 :axisymmetric bubbles, For Ca > 0.54 : bubbles moved faster than liquid and complete by-pass was In uniform film thickness region: realized Ca = Numerical For 0.16, 0.54, and 1 Square GL and [145] ∆p = − 1 Rs∞ Ca < 0.54 : bubbles We  1.0 Analytical moved slower than liquid and reverse flow occurred For Ca = 0.54 : bubbles traveled as the same velocity as the liquid and no stagnation pressure The pressure drop of two-phase flow was greater than single-phase flow The Lockhart and Martinelli model did dH= 0.2 and 0.525 mm − 1 not adequately predict 0.001 ≤ Ca ≤ 0.2 α 2 Square GL Experimental [72] ∆p =∆pl l the pressure drop for 0.004 ≤ Vg ≤ 11 the laminar flow regime 0.001 ≤ Vl ≤ 0.2 [193] The viscosity of the liquid phase had a key role in the frictional part of the pressure loss  1  d = 2 mm = + d Re
3 Circular Experimental ∆p 16 1 0.17 L Ca Pressure drop over the ≤ ≤ s 0.002 Ca 0.04 and GL [44,53]   0.04 ≤ V ≤ 0.3 and 1 slugs l Square Numerical ∆p = 14.2 1 + 0.17 d Re
3 Re ≤ 900 Ls Ca Processes 2021, 9, 870 30 of 40

Table 9. Cont.

Correlation Comment(s) Flow Parameter Cross-Section Phases Method Reference 0.2 mm ≤ w ≤ A decrease in the 0.4 mm h = 0.15 mm channel curve radius −3 enhanced the pressure 4.7 × 10 ≤ −3 drop Ca ≤ 7.4 × 10 Experimental QlµlLg An increase in vorticity 11.6 ≤ Re ≤ 22.6 Rectangular GL [60,183] ∆p = 32 3 3 and hf enhanced the pressure 20.4 × 10 ≤ Numerical linearly Pe ≤ 39.6 × 103 −1 Pressure drop was 0.01 ms ≤ Vl ≤ decreased with 0.07 ms−1 −1 Vg = 0.08 ms The pressure distribution in liquid slug and gas bubble d = 0.5 mm was significantly V = 0.245 ms−1  h i g 4L 1
2 different due to the −1 2D GL ∆p = ε f ρl Vl+Vg Vl = 0.255 ms Numerical [152] l d 2 interfacial effects = The interfacial pressure Re 280 Ca = 0.006 difference at the nose was greater than the tail of the gas bubble The pressure drop behavior with 1 L∗(Ca/Re) 3 was approximated by two asymptotes Total pressure drop The total pressure drop 2 was a summation of = + 7.16(3Ca) 3 d = 1 mm ∆p 16 2L∗Ca Poiseuille flow and Curve fitting after scaled using Taylor components 1.58 ≤ Re ≤ 1024 ≤ ≤ Circular GL Experimental [54] 1 Poiseuille flow was 0.0023 Ca 0.2 Ca
3 0.034 ≤ Bo ≤ 0.28 Re dominant when 1 1 1.92 Re
3 ∗ 3 ∆p = 16+ ∗ L (Ca/Re) was L Ca greater than 0.1 and Taylor flow was dominant for 1 L∗(Ca/Re) 3 less than 0.1 Shorter liquid slugs and higher inertia force empowered the inner recirculation region " 1 #   3 Ls −1.6 dH= 0.4 mm  d  ρ dσ At −≤53,L and Square GL Experimental [184] ∆p = 56.9 1 + 0.07 H l + 5.5 × D10h s 0.05 ≤ We ≤ 5.5 Ls µ2 d Re l Re=200, the pressureH drop approached to that proposed by Kreutzer et al. [44] Pressure drop was assumed as a summation of frictional and interfacial d = 0.25 mm   0.0023 ≤ Ca ≤ ∆p = 16 1 + 0.931 RF 1 components Circular GL Experimental [55] Vl 1 0.0088 Ca 3 +3.34Ca The accuracy of the 41 ≤ Re ≤ 159 proposed pressure drop model increased for Re greater than 150 Pressure drop was assumed as a α 4V µ (1−α)L summation of frictional ∆p = 4Vs L + m c + sf 0.5(R−δ)2 0.5 R2 and interfacial µ 0.248 mm ≤ d ≤ d components L σ 2 0.498 mm 7.446 Ca 3 Stagnant film (sf) −1 Luc R 0.03 ms ≤ Vs ≤ Circular LL Experimental [56] ∆p = 4VsαL + reduced the channel −1 mf R2−(R−δ)2 0.5(R−δ)2 0.5 ms + diameter effectively Qo µc µd 0.1 ≤ ≤ 4 Moving film (mf) Qa 4V µ (1−α)L σ 2 m c + L 7.446 Ca 3 0.5 R2 Luc R considered the dispersed phase frictional drop Processes 2021, 9, 870 31 of 40

Table 9. Cont.

Correlation Comment(s) Flow Parameter Cross-Section Phases Method Reference The dominant role of ∆p = 2Rb w Lc σ ∆F R2 Us Luc the number of bubbles −4 −3 Analytical compared the gas 3.04 × 10 < Ca < 9.6 × 10 GL and ∆ptrans= −aNucLs Circular and [183] 0.053 < εl < 0.917 LL n 1 o holdup on the pressure Experimental 1+ [1 − exp(−bLs)] 2bLs losses The pressure drop prediction was only µ valid for long droplets ∆p = 4 cVmLs L + uc 0.5 R2 Luc The interfacial pressure d = 1.06 mm µ Experimental 4Vm cLf L ≤ ≤   4  L drop increased as the 0.0045 Ca 2 + Ri 1 − uc Circular LL and [48] 0.5 R 1 R ( γ 1) diameter decreases 0.0089 21 ≤ Re ≤ 41 Numerical σ 2 The interfacial pressure + L 9.402 Ca 3 Luc R drop was negligible for very long slugs and droplets Total pressure drop was increased as the total volumetric flow rate = enhances d 1.59 mm 1.45 ≤ Re ≤ " The total# pressure drop   1 567.59 GL RF 1 Qdapproaches3 to an −5 Circular Experimental [57] ∆p = 16 1 + 1.061 V 1 1− Q 4.49 × 10 ≤ LL l Ca 3 +3.34Ca asymptotict value of 16 Ca ≤ 0.067 µc for single-phase flow for 1.5 ≤ µ ≤ 953 1 d Lc 3 d (Ca/Re) greater than 0.12

Besides, Song et al. [180] expressed the total pressure drop considering the total number of water slugs (n), including the pinching off slug,

∆pt = ∆pg + ∆pcr + (n − 1)∆ps (18)

where the first term indicates the pressure drop of single-phase flow of gas, the second term is the critical pressure drop to break up the water slugs, and the last term denotes the pressure drop due to the moving slugs through the capillary. According to the literature, a circular cross-sectional area has been assumed for the gas bubble, where the velocity gradient is limited to the confined liquid region surrounding the bubbles, not inside the bubbles [29]. However, the linear ratio between the pressure and velocity fields is valid for a straight tube, independent of the diameter of the channel. Introducing gas bubbles into the continuous phase flow changes the relation to be nonlinear. The moving bubbles in a tube leave a backfill of liquid left behind the bubble, where the volumetric flow rate of this backflow can be calculated by πrδVg. Marchessault and Ma- son [29] also showed the pressure gradient as a function of surface tension, film thickness, and the radius of the tube. They experimentally determined that at high velocities the liquid film thickness increases in the direction of the gas-bubble movement, when the profile of the bubble shows a tendency to be conic at the front. Table9 presents the correlations to calculate the pressure drop created by bub- ble/droplet in a microflow. One of the first empirical correlations of pressure loss in a capillary tube for horizontal and vertical configurations was experimentally and analyt- ically obtained by Bretherton [30]. He discovered that the pressure drop was enhanced from a coefficient of 14.894 at the leading to 18.804 at the trailing regions due to the slight difference between the frontal and rear menisci of the bubble. Ratulowski and Chang [35] proposed a correlation for pressure drop over a finite gas bubble approaching Bretherton’s law for low Ca along with two other correlations for a square tube. Using a Finite Element Method (FVM), Giavedoni and Saita [36] determined the pressure gradient between two specific points; the first at the stagnation point of the gas bubble, and the second in the uniform film thickness region. Their results revealed an increasing trend in pressure gradi- ent with the capillary number. They also found that the backflow region disappears when the capillary number is sufficiently high. Interfacial pressure along the free surface on the Processes 2021, 9, 870 32 of 40

rear half of the gas bubble was found by them two years later, where the pressure was significantly decreased because of leaving liquid phase from the uniform film thickness region [38]. Kawaji and Chung [71] found that the shape of the ducts had an impact on the frictional pressure loss or the void fraction. They showed that the total pressure drop was a summation of pressure loss in the single-phase part and the two-phase part of a unit cell. The size impact of the cross-section area of a rectangular microchannel was experimentally carried out by Harirchian and Garimella [104] for a boiling heat transfer situation. A slight decrease in pressure drop was found with a heat flux increase caused by liquid viscosity reduction as the liquid phase temperature increased. They also observed an increase in pressure loss as the cross-sectional area decreased. Niu et al. [105] showed the total pressure drop in a two-phase microflow was the summation of the wall friction and the sudden expansion components showing good agreement when compared with homogenous and separated models.

8. Conclusions In this review paper, the literature of GL and LL two-phase flows through microchan- nels with different cross-sectional areas were discussed. Key results can be summarized in the following way. • The maps of Taylor flow and the transition boundaries between each flow regime have been explained by x-y graphs, where the axes are defined by the superficial phase velocities, dimensionless numbers, and volumetric flow rates. • Most often the gas bubbles and dispersed liquid droplets are surrounded by a thin film of carrying phase, except in non-circular tubes with very high void fraction where the dry-out at the inner surface of the tube has been observed. • There is still no universal agreement to classify all flow regimes due to the experimental apparatus and capturing equipment, which force investigators to name the flow regimes differently. • As more efficient numerical solutions and supercomputers emerge, they are used to solve interesting microflow problems with acceptable accuracy. Since the hydrody- namic of the film thickness is crucial and informative, high-resolution images can be produced via CFD. Meanwhile, there is still room to pay more attention to this region because of the important effect of film thickness on the frictional pressure drop, bubble or droplet profile, and heat and mass transfer. • Contact angle can play a major role in the flow pattern as long as two phases are in contact with the inner surface of the tube, except for Taylor flow is, i.e., gas bubbles or dispersed liquid droplets surrounded by a liquid film. • The transition criterion from one flow regime into another one has been investigated by many researchers, resulting in numerous criteria based on the nature of the flow, experimental approach, and key parameters. • The quantitative attempts to correlate slug length, liquid film thickness (recently summarized by [4] in a comprehensive review paper), and pressure drop have been compared. These correlations include non-dimensional numbers, superficial velocity ratios, volumetric flow rate ratios, void fraction, and other thermophysical characteris- tics of phases.

Author Contributions: Conceptualization, A.E.; methodology, A.E.; investigation, A.E.; writing— original draft preparation, A.E.; writing—review and editing, K.P. and Y.S.M.; funding acquisition, Y.S.M. and K.P. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grant numbers 200,439 and 209,780. The authors would also like to thank the Memorial University Libraries Open Access Author Fund, who assisted with funding this paper. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Processes 2021, 9, 870 33 of 40

Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

Dimensionless numbers (discussed in the text) Ar Archimedes number Bo Bond number Ca Capillary number Cn Cahn number Eo Eötvös number Fr Froude number La Laplace number N viscosity number Oh Ohnesorge number Re Reynolds number Su Suratman number We Weber number English letters A [m2] area of the channel A [micron] microreactor oscillation in Equation (9) B [lb hr−1 ft−2] mass velocity of the liquid phase C [–] Chisholm constant (5, 10, 15, and 20 for different flow regimes) d [m] tube diameter f [–] friction factor and a function defined by Wang et al. [179] f0 [kHz] constant vibrating frequency fα [–] a function defined by Wang et al. [179] F [s−1] frequency of gas bubbles g [m s−2] gravity acceleration G [lb hr−1 ft−2] mass velocity of the gas phase h [mm] the height of the rectangular cross-sectional area channel H [m] Kelvin-Helmholtz instability index K [–] end effect of a bubble j [ms−1] superficial velocity in Figure6 L [m] length . m [kg s−1] mass flow rate n [–] number of slugs p [Nm−2] pressure Q [ml s−1] volumetric flow rate r [m] radial distance from the axis of the tube R [m] radius of channel, equivalent radius in Equation (6) s [–] channel gap S [–] slip ratio t [s] time U [ms−1] average velocity v [volt] voltage V [ms−1] superficial velocity w [mm] the width of the rectangular channel x [–] the quality or the dryness fraction X [–] Lockhart-Martinelli parameter z [m] axial length (m) Greek letters α [–] void fraction β [–] the volumetric quality γ [–] the ratio between the dynamic viscosity of droplet to continuous phase δ [m] liquid film thickness ε [–] the gas phase holdup  ρ 0.5 λ = g 62.3 0.075 ρl µ [Pa.s] dynamic viscosity ν the volume of liquid flowing per bubble and slug ξ [m] interface width ρ [kgm−3] density σ [Nm−1] interfacial tension −1 ω0 [s ] angular frequency ∀ [mm3] volume of the dispersed phase ϕ [–] two-phase friction multiplier 1 ψ   2 3 = 73 µ + 62.3 σ l ρl ∆ [–] gradient operator Processes 2021, 9, 870 34 of 40

Subscripts and superscripts * minimum or maximum, and dimensionless parameters 0 initial 3D droplet at the junction a aqueous phase b the equivalent diameter of a sphere b bubble c continuous ca capillary cap head or rear meniscuses of the gas bubble CH1 the first detection point CH2 the second detection point cr critical d dispersed D diagonal direction in Figure 10 f film g gas H hydraulic i interfacial l liquid L lateral direction in Figure 10 ks kerosene-superficial m mixture or mean max maximum mf moving film min minimum o organic phase rec receding s slug sf stagnant film s∞ the radius of the gas bubble in the uniform film thickness region t total th macro-to-micro-scale threshold tp two-phase velocity uc unit cell ws water-superficial Acronyms FEP fluorinated ethylene propylene GL gas–liquid LBM lattice Boltzmann method LL liquid–liquid PDF probability density function VoF volume of fluid XFVM extended finite volume method µ-PIV micro-particle image velocimetry

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