Talimi 2012 International Journal of Multiphase Flow

8/12/2019 Talimi 2012 International Journal of Multiphase Flow

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A review on numerical studies of slug flow hydrodynamics and heat transfer

in microtubes and microchannels

V. Talimi a,, Y.S. Muzychka a, S. Kocabiyik b

a Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. Johns, NL, Canada A1B 3X5b Department of Mathematics and Statistics, Memorial University of Newfoundland, St. Johns, NL, Canada A1C 5S7

a r t i c l e i n f o

Article history:

Received 18 November 2010

Received in revised form 6 September 2011

Accepted 14 October 2011

Available online 21 October 2011

Keywords:

Numerical simulation

Slug flow

Two-phase flow

Microtubes

Microchannels

Pressure drop

Heat transfer

a b s t r a c t

Numerical studies on the hydrodynamic and heat transfer characteristics of two-phase flows in small

tubes and channels are reviewed. These flows are non-boiling gasliquid and liquidliquid slug flows.

The review begins with some general notes and important details of numerical simulation setups. The

review is then categorized into two groups of studies: circular and non-circular channels. Different

aspects such as slug formation, slug shape, flow pattern, pressure drop and heat transfer are of interest.

Theprimary purpose of the present review is to report the existing numerical studies in order to highlight

the research gaps and suggestions for the future numerical simulations. Judging and/or comparing the

different works and correlations can be conducted separately for each of the sections reported here.

According to this review, there are some large gaps in the research literature, including pressure drop

and heat transfer in liquidliquid slug flows. Gaps in research are also found in applications of non-cir-

cular ducts, pressure drop and heat transfer in meandering microtubes and microchannels for both gas

liquid and liquidliquid two-phase flows.

2011 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

1.1. Dimensionless groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

1.2. Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2. Notes on important numerical simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.1. Computational domain extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.1.1. Fixed frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.1.2. Two-phase moving frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.1.3. Single-phase moving frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.2. Two-dimensional versus three-dimensional simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.3. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.3.1. Inlet and exit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.3.2. Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.4. Grid resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.5. Transient time step settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3. Gas/liquid flow in circular microtubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1. Slug formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1.1. T-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1.2. Y-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1.3. Annular (concentric) nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1.4. Pre-mixed inlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.2. Film thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3. Flow patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.4. Bubble shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

0301-9322/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijmultiphaseflow.2011.10.005

Corresponding author.

E-mail address: [email protected](V. Talimi).

International Journal of Multiphase Flow 39 (2012) 88104

Contents lists available at SciVerse ScienceDirect

International Journal of Multiphase Flow

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3.5. Pressure drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.6. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4. Gas/liquid flow in non-circular microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.1. Slug formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2. Film thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3. Velocity and flow patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4. Bubble and interface shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.5. Pressure drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.6. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005. Liquid/liquid flow in circular microtubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1. Slug formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2. Flow patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6. Liquid/liquid flow in non-circular microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.1. Slug formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2. Flow patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.1. Flow pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2. Slug formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3. Film thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.4. Pressure drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.5. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.6. Curved microchannels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Acknowledgment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

1. Introduction

Enhancement in heat transfer at small scales is an active area of

study followed by many researchers. Use of non-boiling two-phase

flows including plug/slug flows or Taylor flows (Taylor, 1961) has

become one of the interesting methods for achieving higher cool-

ing capacities along with other methods including phase-change

phenomena. This type of flow appears as consequent slugs of

two phases which could be gas and liquid or two immiscible liq-

uids (Taylor, 1961). Two effects have been introduced as responsi-

ble factors in enhancement of heat transfer rate in non-boilingtwo-phase flows: internal circulations in the slugs, leading to a

greater radial heat transfer rate, and increased slug velocity when

compared with the same single phase mass flow rate, which is a

consequence of reduced phase fraction.Muzychka et al. (2011a)

showed recently that only the first one is valid. Fig. 1shows the

internal circulations caused by shear forces in the moving plugs.

Most of studies in this area are experimental. As a result of this

and many other reasons, there are significant gaps in areas ana-

lyzed using numerical simulations. One of the most important rea-

sons can be the time-expense of the numerical methods.

Numerical simulation as a method for predicting the heat andmass

transfer characteristics has gained popularity due in part to the

developments in the computational hardware and software.

Although some reviews of two-phase flow studies at microscales

exist in the literature (Shao et al., 2008; Kreutzer et al., 2005a;

Angeli and Gavriilidis, 2008; Gupta et al., 2010a), there is a further

need for a review of numerical works done in this field, in order to

assist researchers setting up their numerical simulations and

finding gaps in numerical studies. The targets of the present review

are: (1) to address the challenges and hardships of numerical sim-

ulations of two-phase slug flows in microscale, (2) to report keyfindings of the existing numerical research, and (3) to introduce re-

search gaps which could be pursued numerically by researchers in

the future. The authors will not focus on the proposed correlations

as this can be found in the existing reviews or requires a separate

review. The present paper is a review on the numerical studies on

hydrodynamics and heat transfer in non-boiling two-phase flows

in small (mini and micro) scales. Much of the work conducted in

recent years has been reviewed and the results gathered and cate-

gorized into two major sections: gas/liquid and liquid/liquid two-

phase flows. These sections are categorized further into two sub-

categories including circular or non-circular channels. While from

numerical point of view there is no difference between the gas/li-

quid and liquid/liquid two-phase flow simulations, role of the gas

phase in heat transfer is mostly ignored due to its much lower heatcapacity. This may further leads to another simplification i.e. using

a single-phase moving frame technique in the simulations which

will be discussed in the following sections. In the case of using a

liquid/liquid two-phase flow for heat removal applications both

of the phases are playing significant roles.

1.1. Dimensionless groups

A significant number of dimensionless groups exist to assist

two-phase flow researchers developing more general results and

models. Reynolds numberRe, Capillary numberCa, Weber number

We, Nusselt number Nu, and Prandtl number Pr, are a few exam-

ples. Only a brief description would be presented here. More de-

tails of definitions and physical interpretations can be found inliterature (Muzychka et al., 2011b).

Fig. 1. Internal liquid plug circulation (a) hydrophobic surface and (b) hydrophilicsurface.

V. Talimi et al. / International Journal of Multiphase Flow 39 (2012) 88104 89


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The Reynolds number,Re, is traditionally defined as:

Re qUDhl

1

and shows the ratio between inertial and shear stress forces. In the

above equation q is the fluid density, l is the fluids viscosity, Uisvelocity, andDh is the hydraulic diameter which is identical to the

diameter in circular tubes. One of the initial and important stepsin numerical simulation of a fluid flow is to examine whether lam-

inar or turbulent conditions exist. Re is mostly used for this deci-

sion. The critical Re corresponding to the onset of turbulence is

2300 for single phase flows in tubes with diameters as small as

50 lm(Incropera et al., 2007). The criticalRe has been reported to

be of 1000 for single moving droplets (Ren et al., 2002) which

could be more applicable in non-boiling two-phase slug flows in

microscale.

The Capillary number,Ca, is defined as follows:

Ca lUr

2

and is the ratio between viscous and surface tension forces. In the

above equation r is the surface tension between the two phases,which cause a Capillary pressure difference between the two phases

at curved interfaces. A combination of Ca, Re, slug length, and

hydraulic diameter of the microchannel make a very important

dimensionless group in two-phase slug flow pressure drop analyz-

ing. Further details can be found in (Muzychka et al., 2011b;

Kreutzer et al., 2005b; Walsh et al., 2009).

The Weber number,We, is defined as:

We qU2Dhr

3

and is the ratio between inertial and surface tension forces. In mini

and micro scales the interaction between two fluids at their inter-

face can be important. The importance of surface tension effects isdetermined based onRe, Ca, and/orWe. The quantity of interest is

Ca or We for lowRe and highRe flows, respectively.

The Prandtl number, Pr, is defined as follow:

PrtalCpk

4

and is the rate of momentum diffusion versus the rate of thermal

diffusion. In the above equation,t is the momentum diffusion, ais the thermal diffusion, Cp is the specific heat capacity, and k is

the thermal conductivity.

The Nusselt number, Nu, is defined as:

Nu

hDh

k

qDhkTw Tm 5

and is a dimensionless heat transfer rate, defined based on the dif-

ference between wall temperature, Tw, and fluid bulk temperature,

Tm. In the above equationh is convective heat transfer coefficient,

and q is heat flux. Another approach exists for defining a dimension-

less heat transfer, which is based on the difference between wall

temperature and inlet temperature, leading to q as follow:

q qDhkTw Ti

6

whereTi is the inlet temperature. The first approach is more appli-

cable in heat exchangers and the second one is useful in heat sinks.

More details are available inMuzychka et al. (2011a) and Shah andLondon (1978).

1.2. Governing equations

The commonly solved equations are for incompressible flows,

i.e. the continuity equation, the momentum equation and the en-

ergy equation (Versteeg and Malalasekera, 1995):

r U!

0 7

qDU!

Dt q~grP lr2U! F! 8

qCpDT

Dt kr2T _s 9

whereP,g,T, _s, andU are pressure, gravitational acceleration, tem-

perature, heat generation, and velocity field, respectively. In the

equations aboveq,l, Cp, andk are density, viscosity, specific heatcapacity, and thermal conductivity, respectively. F is the surface

tension force which needs special treatments to be computed. More

details can be found in Brackbill et al. (1992) and Chen and Li

(1998).

In order to simulate the interface between two phases there are

two major methods applied in the solution: the Volume of Fluid

(VOF) method and the Level Set method (LS). The VOF method is

based on the idea of fraction function C which has a value be-

tween 0 and 1. Basically, Cis one when the cell is full of the main

phase, C is zero if the cell is full of the secondary phase, and

0 < C < 1 when the interface of two phases exists in the cell. The

equation to be solved is (Hirt and Nichols, 1981):

@C

@t UrC 0 10

In the LS method, the interface between immiscible fluids is

represented by a continuous function U. This function represents

the distance to the interface, and equals to zero on the interface,

has a positive value on one side of the interface, and a negative va-

lue on the other side. This way both fluids are identified, such that

the location of the physical interface is associated with the zero le-

vel. The LS equation is as follows (Sussmann et al., 1994):

@U

@t UrU 0 11

Material properties such as the density, the viscosity, the heat

capacity and the thermal conductivity are updated locally based

onCor U. The advantage of using VOF and LS methods are more

precise satisfying the continuity equation and anticipating sharp

interface, respectively, as reported byCarlson et al. (2008). Select-

ing between the aforementioned methods depends also on the

software availability, for example ANSYS Fluent implements the

VOF method.

While numerical codes are still being developed, some commer-

cial codes are been used by a large number of researchers as well.

ANSYS Fluent, CFX, Flow 3D, TransAT, CFD-ACE+, OpenFoam, andCOMSOL Multiphysics are a few examples. Glatzel et al. (2008)

compared the abilities of a number of the aforementioned codes

in microfluidic applications including two-phase flows. Gupta

et al. (2010b)compared the performance of VOF and LS methods

implemented in ANSYS Fluent and TransAT, and reported both

codes are capable of predicting the interface shape in Taylor flow

regime. Carlson et al. (2008) reported some differences between

the numerical results obtained by Fluent and TransAT in the cases

of bubbly and slug flows inside micro tubes. They pointed out that

recirculation flow anticipated by TransAT was not shown by Flu-

ent. As they reported, Fluent needs higher mesh resolution near

the wall in order to capture the liquid film around the gas phase.

Similar results were reported by Gupta et al. (2010b). Krishnan

et al. (2010) compared the different discretization schemes includ-ing first order upwind, second order upwind, power law, QUICK,

90 V. Talimi et al. / International Journal of Multiphase Flow 39 (2012) 88104


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and third order MUSCL in VOF two-phase simulation using Fluent.

They reported better performance of first order upwind and power

law schemes among other schemes. Additionally, they showed that

the node based gradient calculation scheme gives more accurate

results than the cell based scheme.

2. Notes on important numerical simulation details

2.1. Computational domain extensions

Computational domains commonly used by researchers can be

categorized into two main groups: fixed frames and moving

frames. The numerical simulations using moving frame computa-

tional domains are also on two types: two-phase moving frame

simulations and single-phase moving frame simulations. These

types of computational domains are briefly reviewed in this

section.

2.1.1. Fixed frame

A fixed frame computational domain generally includes a junc-

tion and a part of microtube or microchannel after the junction.

Fig. 2 shows this type of computational domain. The two phasesmeet at the junction and form a two phase flow afterward. Many

researchers have used the first type of computational domains to

simulate two-phase flows in microtubes and microchannels (Shao

et al., 2008; Qian and Lawal, 2006; Chen et al., 2009; Kashid et al.,

2007a; Liu and Wang, 2008). They mainly set up the computational

domain as showed in Fig. 2with some differences. In this way, one

should consider the minimum ratio of length to diameter (L/D) or

length to width (L/W) of domain to let the slugs be formed and

the flow become fully developed. This ratio should be at least 40

according to the various reports (Lakehal et al., 2008). This long do-

main requires a large number of iterations especially when the

heat transfer is of interest, therefore the CPU time will be high.

Note that the domain showed in Fig. 2 is typical and researchers

may use it with some differences. For example the geometry ofjunction could be a T-junction (Wu et al., 2008; Yu et al., 2007),

Y-junction (Kashid et al., 2007a; Kumar et al., 2007), concentric

nozzles (Shao et al., 2008; Chen et al., 2009; Gupta et al., 2009 )

or even pre-mixed inflow (Kumar et al., 2007). An example of con-

centric nozzles is showed inFig. 3. Furthermore some researchers

used three-dimensional domains (Liu and Wang, 2008; Ghidersa

et al., 2004) or curved domains (Kumar et al., 2007). The length

of two-phase formation region showed inFig. 2depends on many

conditions like the type of the two-phase flow. This length is re-

ported up to four diameters for bubbly flow and seven diameters

for slug flow (Lakehal et al., 2008).

2.1.2. Two-phase moving frame

In the second type of computational domains only one cell isconsidered and the computational domain is so smaller than the

fixed frame domains. The unit cell generally includes one slug

and two half slugs before and after. Fig. 4a shows a two-phase

moving frame computational domain which has been widely used

to analyze steady state slug or bubble shape and film thickness.

Some researchers have used this type of computational domains

to pursue a particular part of the flow in order to increase the ease

of simulation and simulation time (Muradoglu et al., 2007; Worner

et al., 2007; Taha and Cui, 2006a,b; Fries and von Rohr, 2009 ).

2.1.3. Single-phase moving frame

This type of computational domains has been mostly used for

gas/liquid two phase flows includes only the liquid slug and fixed

(or flexible in some rare research for bubble shape e.g. Kreutzer

et al. (2005b)i.e. moving mesh technique) interfaces. The liquid

gassolid contact angle can be controlled by interface shape.

Fig. 4b schematically shows this type of computational domains.

Mostly a zero shear stress has been applied at the interfaces since

the gas phase viscosity is much less than the liquid phase viscosity.

Single-phase moving frame is a simplified computational domainwhich gives faster results and could be appropriate when there is

no liquid film around the bubbles or dry-out condition. This can

be the case for hydrophobic microchannel materials. In this case

there is no convective heat or mass transfer between the neighbor

Fig. 2. A typical computational domain frequently used in fixed frame simulations.

Fig. 3. Application of fixed frame simulations and concentric nozzles (Gupta et al.,

2009).

Fig. 4. A typical computational domain frequently used in moving frame simula-

tions (a) with film (two-phase) and (b) without film (single-phase).



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liquid slugs through liquid film and all the liquid is circulating

inside the slugs. Baird (2008), Young and Mohseni (2008), and

Talimi et al. (2011a) have used the single phase moving frame

simulations to analyze convective heat transfer in discrete

droplets. Rosengarten et al. (2006) showed that contact angle plays

a significant role on the plug dynamics and whether or not the dry-

out condition appears. The domain presented in Fig. 4b is typical of

many studies and may used in different ways, including three-

dimensional domains (Ghidersa et al., 2004; Worner et al., 2007)

and curved paths (Fries and von Rohr, 2009). Fig. 5 shows examples

of application of the two-phase and single-phase moving frame

computational domains.

2.2. Two-dimensional versus three-dimensional simulation

When a two-phase slug flow in a circular microtube is of inter-

est, one can use a two-dimensional computational domain with

one boundary as an axis. This is applicable in straight microtubes

without any bends and curvature. The 2D assumption is not appro-

priate if a junction is going to be used to generate slugs in circular

microtubes. Furthermore, some researchers have used 2D compu-tational domains for two-phase flows inside square or rectangular

microchannels (for exampleRaimondi and Prat, 2011). Using a 2D

computational domain with or without a symmetry line means a

flow between two parallel plates or through a rectangular micro-

channel with a very large aspect ratio. Talimi et al. (2011b) focused

on the differences between 2D and 3D heat transfer simulation re-

sults for a moving droplet between two parallel plates and dis-

cussed that in some cases care must be taken when using 2D

simulations. Goel and Buwa (2009)showed that even for a circular

microtube, axisymetric simulations will not give same results com-

pared to 3D simulation when bubble generation is of interest in

gas/liquid two-phase flows.Cherlo et al. (2010) compared 2D and

3D simulation results for generated slug lengths in a liquid/liquid

two-phase flow in a rectangular microchannel and showed that

they are comparable. Generally, more computations are required

when a 3D simulation is performed. Adding this to a very time con-

sume nature of two-phase flow simulations, researchers may still

be interested in using 2D computational domains. This issue may

be less important by developing computers in the future.

2.3. Boundary conditions

In this section some important notes on inlets, walls, and exit

boundary conditions have been reported. These notes might be

considered when setting a numerical simulation.

2.3.1. Inlet and exit

The mostly used boundary condition at the inlet is specified

velocity. When constant velocity is considered along with simulat-

ing a junction, care should be taken on the length of each phase

path before the junction in order to maintain fully developed

velocity profile.

While using a specified pressure as the exit boundary condition

has been suggested for these numerical simulations (in which

there is not a fully developed flow which is the case in two-phase

flows because the phases alternate and the circulations inside the

slugs) some researchers recently reported reliable results using

outflow boundary condition (ANSYS Fluent) at the exit (Cherlo

et al., 2010; Goel and Buwa, 2009).

Periodic boundary conditions can be applied at inlet and outlet

with some limitations when using two-phase moving frame com-

putational domains (Shao et al., 2010; Van Baten and Krishna,

2005). For example if heat transfer (or mass transfer) is desired,

then pressure and enthalpy is not the same at inlet and outlet

due to wall shear stress and heat receiving, respectively.

2.3.2. Walls

At microscales, the no-slip boundary condition can be applied in

most of the situations specially when there is a liquid flow inside

microchannels. Deciding on slip or no-slip boundary condition de-

pends on a dimensionless group which is called Knudsen number,

Kn, and defined as follows:

Kn k

L 12

wherek is the mean free path of the gas molecules and L is a char-

acteristic length e.g. hydraulic diameter. However, a slip condition,

Navier slip condition (for exampleChen et al., 2009) can be applied

in order to avoid numerical clutches where there is a moving con-

tact line. More fundamental details can be found in Renardy et al.

(2001) and Spelt (2005).

2.4. Grid resolution

In two-phase numerical simulations there is a need to a very

fine mesh near the walls in the computational domain in order

to get accurate shear stress distribution on the wall and heat trans-

fer from the walls to the flow or capturing the thin liquid film

around the bubbles. A coarse mesh cannot predict the thin film

in hydrophilic microchannels and consequently failed to give a va-

lid shear stress and heat transfer distribution. Even if both phases

touch the wall i.e. a dry-out condition, a fine mesh is needed to

simulate interface moving and/or the shear stress at the corners

if a single-phase moving frame technique is used. Gupta et al.

(2009) discussed this in detail and suggest a criterion for mesh

refinement near the wall based on film thickness which is a func-

tion of Capillary number. Some numerical simulations using coarsemesh can be found in the literature as well. This could be accepted

for purposes other than what mentioned above e.g. slug length.

Qian and Lawal (2006)is an example.

Talimi et al. (2011a)reported that very fine mesh is required at

the corners in order to reach grid independent results when the

wall shear stress is of interest in single-phase moving frame simu-

lations. The grid refinement at the corner has been reported to

about 0.066% of the channel spacing in their study due to high

shear stress gradients at the corners.

More recently, Mehdizadeh et al. (2011)used a dynamic grid

refinement in order to avoid using a very high resolution mesh

and this way they approached valid results with a less number of

grids which means a faster numerical simulation. In their work,

they refined the grid in the vicinity of the interface and coarsenedthe grid elsewhere where there is only one phase and no need to a

Fig. 5. Application of moving frame simulations: (a)with film(Liuand Wang, 2008)and (b) without film (Sammarco and Burns, 2000).



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high resolution mesh. Since numerical simulation of two-phase

slug flow is a transient simulation in nature and the interface

moves with the slugs, there is a need to update the grid (coarsen

and refinement) after a few time steps.

2.5. Transient time step settings

A very important step in transient numerical simulation is time

step sizing. The time step can be adjusted using a dimensionless

group i.e. Courant number which is defined as follows:

Co u Dt

Dx 13

and is a comparison between the particle moving distance during

the assumed time step and control volume dimension. A high Co va-

lue leads to an unstable numeric approach and a low Co value

means a small time step size and consequently a large simulation

time, so there is a need to optimize Co using appropriate time step

size. Furthermore, as the mesh becomes finer the time step should

be decreased as well in order to hold Co in its safe range. A typical

time step order of magnitude of 1 105 (s) or 1 106 (s) has

been used by the researchers (see the three-dimensional simulation

performed byCherlo et al. (2010)for example).A wiser time step adjustment is using a variable time step by

implementing a fixed Courant number which is available in ANSYS

Fluent (for exampleGupta et al., 2009). In this method, the time

step is being modified based on the critical cells size and local

velocity components to hold the maximum Courant number to a

fixed value.

3. Gas/liquid flow in circular microtubes

Most of the numerical studies on two-phase flows in micro-

channels are conducted for gas/liquid flows in circular microtubes.

In order to report these studies, this field is categorized into several

sections including slug formation, film thickness, flow patterns,

bubble shape, pressure drop, and heat transfer.

3.1. Slug formation

The process of slug formation in circular channels has been re-

ported as a strong function of type of mixing zone and studied for

different mixing zone shapes: T-junctions (Qian and Lawal, 2006;

Kumar et al., 2007; Dai et al., 2009), Y-junctions (Kumar et al.,

2007; Ilnicki et al., 2009), pre-mixed inlets (Qian and Lawal,

2006; Kumar et al., 2007), and annular nozzles (Shao et al., 2008;

Gupta et al., 2010b; Chen et al., 2009; Goel and Buwa, 2009;

Narayanan and Lakehal, 2008). The parameters of interest which

have been studied by researchers are diameter of the mixing zone,

channel curvature (for meandered channels), and superficial veloc-

ities of the two phases. The forces which have been studied by re-cent researchers include surface tension, viscous friction, inertial,

and gravitational forces. The effects of these forces can be pre-

sented using the different dimensionless groups such as the Rey-

nolds number, Re, and Capillary number, Ca.

Wall adhesion and contact angle play a prominent role in slug/

bubble formation as reported in some recent literatures. For exam-

pleCherlo et al. (2010)reported an increase in slug length with a

decrease in contact angle. On the other hand, Gupta et al. (2009)

reported that the wall contact angle plays no role when the mesh

is fine enough near the walls because the thin liquid film around

the bubbles can be captured using this fine mesh. This could be

true when the channel material is hydrophilic and there is no

gas/liquid/solid contact angle. In the case of hydrophobic materials,

the wall contact angle is still important. However the mesh resolu-tion in the numerical simulations conducted by Cherlo et al. (2010)

is much coarser than whatGupta et al. (2009)used. This seems to

be unavoidable (especially in 3D cases) since using fine mesh like

whatGupta et al. (2009)suggested is very time consuming.

3.1.1. T-junction

Dai et al. (2009) performed a highly detailed numerical study on

slug formation in T-junction. They showed three stages in slug for-

mation: expansion, collapse, and pinching off. They also arguedthat pressure increase in the liquid phase, shear stress, and surface

tension are three forces which are involved in the process of slug

formation. The different arrangements of liquid and gas inlet direc-

tions and their effects on slug formation were studied numerically

by Qian and Lawal (2006). They showed that in order to achieve

shorter gas slugs, gas and liquid phases should be fed head to head

or perpendicular to each other, while the liquid flows parallel to

the microchannel. Decreasing the diameter of the mixing zone,

leads to a decrease in the gas slug length.

3.1.2. Y-junction

Ilnicki et al. (2009)showed numerically that liquid slug length

increases and gas slug length decreases with a decrease in gas

superficial velocity or an increase in liquid superficial velocity for

a Y-shape microchannel. Similar trend observed by Kumar et al.

(2007). They also reported that slug length ratio has a 2030% dif-

ference compared with the flow rate ratio.

3.1.3. Annular (concentric) nozzles

A three stage process of slug formation using concentric nozzles

consists of expanding, contracting, and necking was reported

by Shao et al. (2008), similar to what observed by Dai et al.

(2009) in T-junction. These stage names refer to the movement

of gasliquid interface at the lower end of the gas bubble near

the nozzle. At the first stage, expanding, the gas/liquid interface

moves toward the microtube wall due to injection of gas. As the

bubble grows to a certain level, the interface movement reverses

and it moves away from the wall toward axis. This is the contract-

ing stage. By injecting more gas, the gas bubble grows further in ra-dial direction and leaves only a thin film of liquid to pass.

Therefore, liquid pressure makes the relation region between gas

bubble and gas inlet stream slender i.e. necking stage.Fig. 6shows

these three stages.

Chen et al. (2009)stated that bubble departure size (maximum

gas slug size just before departure) depends on liquid and gas

superficial velocities. By increasing gas superficial velocity (or

decreasing liquid superficial velocity), the departure size and the

Fig. 6. Slug formation process (Shao et al., 2008).



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length of the gas slug increase. (The same results were also re-

ported byKumar et al. (2007).) They also showed that under the

same liquid and gas superficial velocities (ULand UG) i.e. same flow

conditions, the size of Taylor bubbles formed inside a given geom-

etry of microtube can vary distinctly using different bubble gener-

ation devices. In other words, for a given set ofULand UG, different

flow patterns can occur inside the same channel.

Another detailed numerical study on slug formation process

using concentric nozzles was performed by Goel and Buwa

(2009), in which the effect of different parameters such as gas

and liquid superficial velocities,Ca number, nozzle geometry, con-

tact angles, and physical properties of liquid were of interest. They

showed that wall adhesion is a very important parameter in the

process of slug formation and an increase in wall contact angle

and a decrease in nozzle wall contact angle would decrease the

bubble formation period. Surface tension has a significant effect

on this process and the bubble formation period increases with

an increase in surface tension.

Narayanan and Lakehal (2008) focused on the effect of gravity

on the breakup into slugs and according to their results, the slug

breakup happens slightly earlier for up-flow, which leads to a high-

er breakup frequency.

3.1.4. Pre-mixed inlets

Qian and Lawal (2006) numerically investigated the effect of

pre-mixing level and mixing zone geometry on gas slug length.

They showed as the two phases (gas and liquid) are mixed better

at the inlet of microchannel, the gas slug lengths become shorter

(similar results were also reported byKumar et al. (2007)).Kumar

et al. (2007) also reported that by increasing the curvature ratio

(the ratio of coil diameter to tube diameter) the slug length in-

creases for both gas and liquid phases. They proposed the following

reasons for this behavior. For low curvature ratios small slugs

formed due to strong centrifugal forces. As the curvature ratio in-

creases, the centrifugal force becomes weaker and slug length in-

creases. They also showed that by decreasing Ca, the gas slug

length increases, by showing that when surface tension increases

or viscosity decreases, the gas slugs become longer.

3.2. Film thickness

The effects of different parameters on the liquid film thickness

have been studied. These parameters areCa,Re,Fr, bubble length,

and flow direction (with gravitational effects). Based on Brether-

tons theory (Bretherton, 1961) the ratio of liquid film thickness

around bubbles, to the diameter is only a function ofCa:

d

Dk 0:66Ca2=3 14

where d and Dh are liquid film thickness and hydraulic diameter,respectively.

Gupta et al. (2009)compared the numerically calculated liquid

film thickness with different empirical correlations and showed

that the best agreement is with Bethertons prediction at low Ca

(similar results were reported by Martinez and Udell (1989)).Chen

et al. (2009)calculated the liquid film thickness around the bubble

for Ca 0.006 and 0.008, and compared their results with the

Bretherton correlation. Their results show some differences from

Brethertons prediction. WhileTaha and Cui (2006a) reported the

independence of liquid film thickness around a rising bubble on

the bubble length in vertical microtubes,Chen et al. (2009)argued

since the bubble length in their study was smaller than three times

the microtube diameter, the results show some difference from

Brethertons prediction. Shen and Udell (1985)reported an overprediction of Brethertons theory for Ca higher than 0.005.

Fig. 7 shows the results of numerical study performed by Giave-

doni and Saita (1997) on the liquid film thickness. As one can see in

Fig. 7, Re has some effects on liquidfilm thickness for high values of

Ca.Aussillous and Quere (2000)suggested a correlation for liquid

film thickness at high Ca as a function of Ca and Re or Ca and

We. Edvinsson and Irandoust (1996) (and also Taha and Cui,

2004) showed that the liquid film thickness increases with Ca, Re

andFr

. They argued that for small microchannels it is sufficient

to correlate the liquid film thickness with Ca. As flow rates and

diameters increase, the effects of inertial forces as well as gravita-

tional forces become more pronounced.

Fig. 8a shows the range ofCain which numerical studies on the

liquid film thickness have been conducted. As this figure shows,Ca

lower than 0.001 has been neglected in the numerical simulation.

Assuming water as liquid phase and air as gas phase, this value

represents a superficial velocity of 0.07 m/s. As lower velocities

are applicable in the microfluidics, numerical studies for lowCa

could be performed in future research.

Thickness of the liquid film around the bubbles in a gas/liquid

two-phase flow might affect heat transfer process since it affects

the circulation and bypass flows. As reported by Taylor (1961),

with an increase in film thickness more fluid bypasses through this

film and less fluid circulates inside the liquid slugs. The circulation

inside the slugs is the reason of heat transfer enhancement, as dis-

cussed earlier.

3.3. Flow patterns

Bubble length, Peclet numberPe, Bodenstein numberBo, Capil-

lary numberCa, pressure gradient, flow direction (gravitational ef-

fects), channel curvature (for curved microchannels), and

superficial velocities of the two phases have been reported by

many researchers as the main factors influencing flow patterns

and consequently heat and mass transfer. Another important factor

is whether the bubble touches the channel walls, i.e. dry-out con-

dition, or not.Giavedoni and Saita (1997) and Martinez and Udell (1989) re-

ported different flow patterns for high and low Ca. Based on their

Fig. 7. Liquid film thickness around moving bubbles versus Re and Ca for parallelplates (dashed lines) and circular tubes (solid lines) (Giavedoni and Saita, 1997).



8/17

results, circulation regions inside the liquid slugs become smaller

as Ca increases and showed that these results agree with Taylor

(1961).

Slip ratio between bubbles and liquid has been studied numer-ically (Edvinsson and Irandoust, 1996; Ua-arayaporn et al., 2005;

He et al., 2007).He et al. (2007)found that for Taylor flow when

the wall is perfectly wetted by liquid, the gas bubble to liquid

velocity ratio is approximately 1.2, which agrees well with the Ar-

mand correlation (Armand and Treschev, 1946) but when the gas

bubbles contact with the wall directly i.e. a dry-out patch, the

gas bubble flows with the same velocity as liquid slugs (slip ratio

of 1). Ua-arayaporn et al. (2005) reported higher ratios of gas veloc-

ity to liquid velocity for higher pressure gradients and argued this

is because the change in bubbles shape for higher pressure gradi-

ents. At higher pressure gradients, the bubble is elongated in the

central region of the tube or pipe and less is influenced by the

no-slip wall condition. Edvinsson and Irandoust (1996) argued that

for small microchannels it is sufficient to correlate the relativevelocity of gas plugs with Ca. As flow rates and diameters increase,

the effects of inertial forces as well as gravitational forces become

more pronounced. In addition, the size of the circulation regions

within the liquid plug decreases rapidly with increases in liquid

film thickness. Ilnicki et al. (2009) reported that for high values

of overall velocities (UG+ UL> 0.4 m/s) the time necessary to reach

a steady Taylor flow increases significantly if the slip ratio (UG/UL)

is higher than 1. On the other hand, for the overall velocity lower

than 0.4 m/s this necessary time increases only if the slip ratio is

higher than 2.5.

Axial transport phenomena (considering mass transfer) be-

tween liquid slugs through the liquid filmaround the bubbles have

been studied bySalman et al. (2004) and Muradoglu et al. (2007).

Salman et al. (2004)investigated numerically the effect of axialmixing in liquid phase of Taylor flows for low Bo, where diffusion

is sufficiently large, and reported an increase in axial mixing in li-

quid slugs with increasing Caand argued this is because of a thick-

er liquid film around the bubbles and better communication

between slugs. They also showed that axial mixing increases with

an increase in bubble and slug length and suggested using short

bubble and slug lengths in order to have reduced axial mixing.

Muradoglu et al. (2007) studied the effects ofPe on the axial

mass transfer (dispersion) in the liquid slugs. They found that

convection and molecular diffusion control the axial dispersion

for different Peclet numbers, Pe. They introduced three different re-

gimes ofPe:

(1) Convection-controlled regime whenPe > 103.

(2) Diffusion-controlled regime when Pe < 102.

(3) Transition regime when 102 6 Pe6 103.

Curved microtubes have been studied by Fries and von Rohr

(2009). They showed that the symmetrical velocity profile of

straight microtubes changes to an asymmetrical one for the mean-

dering channel configuration. Therefore radial mass transfer inside

the liquid slug increases and mixing length reduces. A smaller ra-

dius of curvature provides more mixing rather than larger radii.

They argued this is because of larger centrifugal forces. According

to their discussion, the optimal slug length for fast mixing depends

on the bend geometry. While longer slug lengths provide an en-

hanced mass transfer due to the bending angle of the slug, shorter

liquid slugs have smaller turning angles, providing an enhanced

asymmetrical flow pattern.

Taha and Cui (2004) reported that at low Ca, the shape of

streamlines at the front and end of the bubble are almost the same,

bending sharply in the path of circulation. They also mentioned

that this localized effect of the bubble can be seen only within a

bubble diameter from the bubble nose. According to their results,

the circulation region in the liquid slugs becomes smaller and

the center of the circulation zone shifts toward the axis, as Ca in-

creases. Also, by increasingCa, the thickness of liquid film flowing

around the bubble increases, and complete bypass flow occursaroundCa= 0.5. They observed two stagnation points in the liquid

plug: one on the bubble tip and the second inside the liquid plug.

3.4. Bubble shape

Typically, Taylor bubble shape depends on different parameters

such as Ca and Re and has a meniscus front nose and a relatively

flattened bottom. The length of the Taylor bubbles depends

strongly on the mixing configuration. The liquid film formed be-

tween the bubble and the wall is uniform at the middle portion.

The film thickness of the liquid around the gas bubble increases

continuously towards the front nose (as it has been showed sche-

matically inFig. 2), while it shows somewhat wavy behavior to-

wards the bottom portion.Kreutzer et al. (2005b) and Chen et al. (2009)reported a change

in the front and rear shapes of the departed Taylor bubbles with Re

(defined based on the two-phase velocity U= UL+ UG). According to

Chen et al. (2009), at low Re, both the front and rear sides of Taylor

bubbles show a hemispherical cap shape. As Re increases, the cur-

vature radius of the bubble tip become smaller and the rear cap be-

comes flattened (similar results are reported by Kreutzer et al.

(2005b)). The dependency of bubble shape on Re is showed in

Fig. 9.Chen et al. (2009)also showed that when the liquid super-

ficial velocity is small enough as compared to the gas superficial

velocity, long Taylor bubbles can be formed by merging a growing

bubble with the departed Taylor bubble in front of it. This approach

is called Taylor-pairing or doubling.

Edvinsson and Irandoust (1996) showed that as Ca increases,the rear side of the bubble becomes less convex and inverted

Fig. 8. Ca range in which liquidfilm thickness has been studied numerically for gas/

liquid flows: (a) circular microtubes and (b) non-circular microchannels. (see

above-mentioned references for further information.)



9/17

gradually. They also reported that as Reincreases, a multiple wave-

let appears in the film near the rear side of the bubble and its

amplitude increases withRe.

Taha and Cui (2004) reported almost similar curvature of Taylor

bubble ends at low Ca and nearly identical streamlines at both

ends of the bubble. They also showed that as Caincreases, the bub-

ble nose becomes more slender and liquid film flowing around the

bubble is thicker (similar results reported by Martinez and Udell

(1989)). They discussed three separate regions in the leading edge

of the bubble: the bubble cap region, the liquid film region and the

transition region between them. According to their results, in-

crease inCa result in an increase of the liquid film thickness and

the sharpness of the bubble nose. This results in smaller bubble

caps and larger transition regions.

Taha and Cui (2006a) found that by decreasing the Morton

number,Mo, under a constant value of Eotvos number, Eo, the cur-

vature of the bubble nose increases and the bubble tail flattens,

which results in an increment of the liquid film thickness around

the bubble. Also the curvature of bubble nose increases as Eo goes

up.

The length of the Taylor bubbles has been studied numerically

byShao et al. (2008) and Kumar et al. (2007).Shao et al. (2008)re-

ported the bubble size increases with increasing gas and decreas-

ing liquid superficial velocities. Also, bubble size is mainly

affected by surface tension and only slightly by density and viscos-ity. They reported an increase in bubble size with increasing nozzle

size, and argued the reason is due to the decrease in gas flux when

using larger nozzles, while keeping the same gas flow rate. The sur-

face tension force increases as nozzle becomes larger. Former has a

detaching effect and later has an attaching effect, therefore the

bubble formation time and consequently its size increases. A

dependency on bubble length on inlet conditions was also reported

byKumar et al. (2007)for pre-mixed inlet configuration in mean-

dering microtubes.

Ua-arayaporn et al. (2005)studied the effect of pressure gradi-

ent on the shape of bubbles in a slug flow through microtubes and

found that the bubbles are nearly spherical under the low pressure

gradients.

3.5. Pressure drop

Pressure drop in two-phase slug flow increases as a result of cir-

culations inside the slugs and the hydrodynamic effects of a two-

dimensional velocity field near the wall. As reported by Kreutzer

et al. (2005b) and Walsh et al. (2009) the ratio betweenCa and

Re is a prominent parameter in pressure drop modeling of gas/li-

quid two-phase slug flow. Pressure drop in two-phase flows inside

circular microchannels has been studied as a function of slug

length, bubble length, bubble frequency, and Ca. Also, some

researchers have noticed the effects of curvature and channel

width for two-phase flows inside meandering microchannels. Ta-

ble 1 shows that most of the numerical studies on pressure drop

conducted for gas/liquid two-phase flows are in circular micro-

channels and other useful shapes have not yet been studied.

Kreutzer et al. (2005b) and Gupta et al. (2009)showed that in a

computational unit cell including a Taylor bubble and two half li-

quid slugs ahead and behind the bubble, the pressure distribution

along the axis in the bubble is constant and higher than that of the

liquid phase. Similar results have also been reported by Taha and

Cui (2004). Kreutzer et al. (2005b) suggested modeling the pres-

sure drop as sum of two elements. Firstly, the pressure drop in

the slug (liquid phase) as a single phase, and secondly, the effects

of gas phase or interface as a function ofRe and Ca. According to

their results, pressure in the liquid slug decreases along the axis,

and an oscillation was found right behind the bubble, seeFig. 10.

Kreutzer et al. (2005b) argued that oscillation happens because

of an inundation on the interface. Gupta et al. (2009) reported a

complex pressure distribution in this region because of numerical

effects.

Chen et al. (2009)showed that pressure drop increases rapidly

with decreasing slug length or increasing bubble length. Therefore,

using different gas injector systems e.g. nozzles, can affect pressure

drop by influencing gas slug length.

Lakehal et al. (2008) reported an increase of 1415% in pressure

drop of slug flow with LGB/L = 5.14 compare with a single phase

flow and suggested an optimal void fraction range from 0.2 to0.36 in order to increase heat transfer without significant increase

in pressure drop and consequently pumping power.

Taha and Cui (2004) observed fluctuations in the wall shear

stress ahead of and behind the slug, and nearly zero pressure drop

in the central region. Based on their results, these fluctuations de-

crease with an increase inCa. They also reported that the pressure

drop across the bubble front increases with increasingCa.

He et al. (2007) and He and Kasagi (2008)stated that increasing

in bubble frequency and decreasing in slug length will increase

pressure drop in both flow conditions of with and without liquid

film. He and Kasagi (2008) reported that at one microtube

diameter away from gas bubble, shear stress is almost equal to

the single-phase Poiseuille flow. In the other words, the increasing

Fig. 9. Taylor bubble shape for Re =1, 10, 100, 200 at Ca=0.04 (Kreutzer et al.,

2005b).

Table 1

Numerical studies on pressure drop in circular microtubes and non-circular microchannels.

Gas/liquid Liquid/liquid

Fixed frame Moving frame Fixed frame Moving frame

Gupta et al. (2009, 2010b) Kreutzer et al. (2005b) Kashid et al. (2010) No studies found

Akbar and Ghiaasiaan (2006) Fries and von Rohr (2009)

Chen et al. (2009) He and Kasagi (2008)

Qian and Lawal (2006) Mehdizadeh et al. (2009)

Mehdizadeh et al. (2009)

Yu et al. (2007) Taha and Cui (2006b) No studies found No studies found

No studies found No studies found No studies found No studies found

No studies found No studies found No studies found No studies found



10/17

effect of two-phase flow on shear stress is significant near the

interface.

Pressure drop in two-phase flow in curved microchannels was

studied byFries and von Rohr (2009). They reported channel cur-

vature and width as effective parameters and stated that as these

parameters decrease, the pressure drop of the gasliquid Taylor

flow increases.

3.6. Heat transfer

Convective heat transfer rate in non-boiling two-phase flowsmay be affected by a large number of issues (Muzychka et al.,

2011a) including velocity and flow patterns, slug length, internal

circulations, phase volume fraction, fluid physical properties,

channel shape, and thermal boundary conditions. Several research-

ers (Gupta et al., 2010b; He et al., 2007; Lakehal et al., 2008;

Narayanan and Lakehal, 2008; Ua-arayaporn et al., 2005) reported

notably higher Nu in two-phase gas/liquid slug flows in microtubes

compared with single phase flows. According to the literature, slug

length and gravity also affects the heat transfer.Table 2illustrates

different numerical studies on heat transfer in gas/liquid two-

phase flows in circular microtubes and briefly shows the simula-tion assumptions and conditions applied by researchers.

Ua-arayaporn et al. (2005) showed that in a slug flow, higher Nu

numbers exists in the region of liquid film around the bubbles.

They argued this is because of the small difference between local

bulk and wall temperatures. Similar results recently have been re-

ported byGupta et al. (2010b) and He et al. (2007) .

According toHe et al. (2007), the local Nu is large in the area

where the bubble exists and that the liquid film between gas bub-

ble and wall significantly increases the local Nu. They also argued

that the circulation within the liquid slug is the primary mecha-

nism of enhancement in the heat transfer.

Gupta et al. (2010b)reported higherNu respect to laminar and

fully developed single phase flows. They showed that using slug

flow,Nuincreases approximately by a factor of 2.5 in both isother-

mal and isoflux boundary conditions. They also found that the gas

phase does not play a significant role in axial heat transfer process

(between gas and liquid slugs). This does not necessarily mean that

the role of liquid film around the bubble in heat transfer is ignor-

able. Based on Fig. 11from Gupta et al. (2010b)almost 40% of heat

transfer in the case of constant wall temperature occurs in the re-

gion of thin liquid film around the bubble.

Fig. 10. Wall pressure in the axial direction (Kreutzer et al., 2005b).

Table 2

Numerical studies on heat transfer in gas/liquid two-phase flows in circular microtubes.

Authors Flow conditions Geometry Research aspects

Ua-arayaporn et al. (2005) 14 < Re< 435 Moving frame Heat transfer for isoflux wall boundary condition

0.4 < We < 1 30 Circular tube

D= 10, 20 lm,L = 2D

Fukagata et al. (2006) 0.0087 < Ca< 0.27 Moving frame Heat transfer for isoflux wall boundary condition

16 < ReL< 490 Circular tube

D= 20 lm,L = 2D

He et al. (2007) Re= 9 0 Moving frame Heat transfer for isoflux wall boundary condition

We= 11 Circular tube

D= 10 lm

Narayanan and Lakehal (2008) Ca= 0.0154 Fixed frame Heat transfer for isothermal wall boundary conditions

We= 17 Circular tube

Re= 1110 D= 1 mm, L = 40D

Lakehal et al. (2008) UG= 0.66m/s Fixed frame Heat transfer for isothermal wall boundary condition in slug and bubbly flows

UL= 1 .11 m /s Circular tube

D= 1 mm, L = 40D

Gupta et al. (2010b) Re= 280 Fixed f rame Heat trans fer f or isothermal and is oflux wall boundary conditions

Ca= 0.006 Circular tube

Waterair,a = 0.51 D= 0.5 mm, L = 40D

Fig. 11. Wall heat flux for isothermal boundary condition. Shaded area shows gas

bubble (Gupta et al., 2010b).



11/17

Lakehal et al. (2008) argued that there are two separate heat

transfer regions in gasliquid slug flow in micro tubes: one right

at the initial breakup and one far downstream in the fully devel-

oped region. In the first region, the local temperature gradient

and heat transfer increases due to compressing thermal boundary

layer by the tale of bubble. In the second region, the fluid flowcoming from liquid film around the bubble into the liquid slug be-

hind the bubble, which was named as a jet-like flow by the

authors, was introduced as the responsible mechanism of convec-

tive heat transfer. According to their results, there is a substantial

increase in Nu with increases in the gas bubble length LGB. For

LGB/L = 5.14 the mean Nu reached a value of 17.20 for the fully

developed region which is much greater than 3.67 for single phase

heat transfer in a microtube having a constant wall temperature

boundary condition.

Narayanan and Lakehal (2008) reported a substantial increase

in heat transfer rate with increasing gas slug length. They obtained

an average Nu of 15 for slug flows and indicated that using slug

flows may lead to an overall enhancement in heat transfer by a fac-

tor between 3 and 4 compared to single-phase flows. They alsoshowed that gravity has some effects on heat transfer in slug flows,

and in particular the down-flow case where the direction of gravity

is the same as direction of flow, the average Nu is about 4% higher

compared to the zero gravity cases.

A number of mass transport studies exist in the literature.

While, most of them studied the mass transfer between slugs i.e.

axial transport, only mass transfer studies between slugs and wallcan be analogous to heat transfer because they are both in radial

direction. Irandoust and Andersson (1989)is an example.

As it is common in approximately all of the numerical studies

that have been reviewed, the fluid properties are assumed to be

constant at an appropriate temperature. Fei et al. (2010) is one of

the rare studies in which two-phase flow was numerically studied

using temperature dependant properties.

4. Gas/liquid flow in non-circular microchannels

Numerical studies on hydrodynamics and heat transfer of gas/

liquid two-phase flows in non-circular microchannels are not as

common as those for circular microtubes. The first issue whichcan be important is the necessity of three dimensional modeling

Table 3

Numerical studies on hydrodynamics and heat transfer in gas/liquid two-phase flows in non-circular microchannels.

Authors Flow conditions Geometry Research aspects

Liu and Wang (2008) 0.003 < Ca < 1 Moving frame Bubble shape, flow patterns, film thickness

Square and equi-triangular ducts

Dh=1 mm, L = 16Dh

Yu et al. (2007) 0.006 < Ca < 0.2 Fixed frame Pressure drop, slug formation, flow patterns, film thickness

Airsilicon oil Square and rectangular ducts

125lm 125 or 250 lm

Ghidersa et al. (2004) Re= 76, 3 Moving frame Bubble shape, film thickness, flow patterns

Ca= 0.043, 0.205 Square duct (3D)

2 mm 2 mm 2 mm

Worner et al. (2007) Ca= 0.2 Moving frame Bubble shape, flow patterns

Square duct (3D)

2 mm 2 mm 2 mm

Taha and Cui (2006b) 0.001 < Ca < 3 .04 Moving frame Bubble shape, pressure drop, flow patterns, film thickness

Square duct (3D)

2 mm 2 mm 22mm

Young and Mohseni

(2008)

Re= 1 00, 400 Moving frame Heat transfer for isothermal and isoflux wall boundary conditions

Single droplet between parallel

plates

0.125 < H/L< 1

Onea et al. (2009) Mo= 0.0004926 Moving frame Bubble shape, film thickness, flow pattern

H= 0 .03, 3 Square and rectangular duct (3D)

Air/methyl chloride-

water

Dh= 2 mm

Oztaskin et al. (2009) 0.11 < Ca < 0.23 Moving frame Velocity and flow pattern, flow stability

Square duct (3D)

2 mm 2 mm

Santos and Kawaji (2010) 0.018 < UG< 0.791 m/s Fixed frame Slug formation, flow pattern

0.042 < UL< 0.757 m/s Square ducts (3D)

Airwater Dh= 113 lm

Oprins et al. (2008) U= 0.1 m /s Moving frame Flow pattern and heat transfer for isothermal wall boundary

condition

Water Single droplet between parallel

plates

L= 2.8 mm, H= 1 mm

Heil (2001) 0.05 < Ca < 5 Moving frame Film thickness, flow pattern0 < Re< 280 2D channel

Giavedoni and Saita

(1997)

0.005 < Ca < 1 Moving frame Film thickness, flow pattern

0 < Re< 7 0 Circular tubes and parallel plates

Guo and Chen (2009) 6.4 104 < Ca < 0.017 Fixed frame Slug formation, flow pattern

Rectangular duct (2D)



12/17

for non-circular geometries, which makes the simulation time

expensive. Parallel plate microchannels are an exception. Table 3

shows the numerical studies on gas/liquid two-phase flows in

non-circular microchannels.

4.1. Slug formation

Slug formation process in non-circular microchannels has beenwidely neglected in the recent numerical simulation. Almost all of

the numerical studies on slug formation process in gas/liquid two-

phase flows have been carried out for circular microtubes. This

may be because of applicability of a two dimensional axisymmetric

geometry. The studies performed bySantos and Kawaji (2010)for

square microchannels and Yu et al. (2007) for square and rectangu-

lar microchannels are exceptions. Based on the results of Santos

and Kawaji (2010) the length of gas slugs increase with an increase

in gas superficial velocity or a decrease in liquid superficial veloc-

ity. Yu et al. (2007) performed a lattice Boltzmann numerical study

on the process of slug formation in non-circular microchannels and

reported a dependency of the process onCa. They showed that for

high Ca, small bubbles formed, with a diameter less than micro-

channel width. So slug flow would not occur. They argued that thisis because of weak surface tension forces that cannot dominate

shear forces. According to their results, gas inlet pressure is higher

than liquid inlet pressure in the expanding stage of slug formation.

When the liquid inlet pressure overcomes the gas pressure, the

second stage starts. The third stage, necking, is when a peak occurs

in the difference between gas and liquid inlet pressures. The afore-

mentioned procedure can be seen in Fig. 12.

Using 2D simulations for rectangular microchannels Guo and

Chen (2009) reported some critical contact angles above which

the Taylor bubble formation will not take place and showed that

this critical contact angle depends on Ca. This shows the important

role of the contact angle (i.e. using different wall materials) in con-

trolling the slug formation process in microchannels. Obviously

more research can give deeper insight on this subject.

4.2. Film thickness

The effects of different parameters includingCa, gravity effects,

channel shape and geometry, and bubble length on the thickness of

liquid film around the bubble has been of interest in recent

research.

Ghidersa et al. (2004) reported that in square microchannels,

the liquid film thickness around the Taylor bubbles increases with

increases in theCa.

Taha and Cui (2006b) showed that there is a difference between

the liquid film thickness for upward and downward flows in square

microchannels, for Ca = 0.009. According to their results, for this

lowCa, the film thickness around the bubbles rising in a micro-

channel (upward flow) is higher than downward flow, and they ar-gued that this is because of gravity forces.

Onea et al. (2009)considered a cylindrical bubble in square and

rectangular minichannels with the same hydraulic diameters and

flow parameters. They showed that the bubble in the square chan-

nel has a circular cross section while in the rectangular channels

the bubble is squeezed between the long walls. Therefore, the

thickness of the liquid film relative to the smaller dimension is

decreasing while the other is increasing.

Liu and Wang (2008)reported a thinner liquid film thickness at

the side walls in equilateral triangular microchannels as compared

with square microchannels for the same hydraulic diameters and

initial volume of Taylor bubbles. They also showed that liquid film

thickness in square microchannels is less than that in circular

microtubes over the full range ofCa . They presented the reasonsas follows: the symmetry of the circular microchannel leads to a

uniform liquid film between Taylor bubbles and the wall, but in

square and equilateral triangular microchannels most of the liquid

is squeezed into the corners which results in thinner liquid films

on side walls.

According to Yu et al. (2007), the liquid film around the bubbles

could become very thin or even partially dry-out in small micro-

channels with width of around 50 lm. They reported that the film

thickness depends on the magnitude of the surface tension, the

bubble length, the bubble velocity and the surface property.

The range ofCa in which numerical studies on the liquid film

thickness have been conducted for non-circular microchannels is

showed inFig. 8b. As can be seen, most of the research was per-

formed for square cross sectional microchannels. Only Yu et al.

(2007) carried out simulation for rectangular microchannels and

only for aspect ratio of 2. Therefore, a gap exists for Ca lower than

0.006 for rectangular microchannels with the aspect ratio of 2, and

also for other aspect ratios in all values ofCa. Similar to rectangles,

the only triangles shape which has been studied numerically was

equi-triangles byLiu and Wang (2008).

4.3. Velocity and flow patterns

According to the published literature, velocity field and flow

patterns of gas/liquid two-phase flow in non-circular microchan-

nels are both affected by Ca, channel shape and geometry, slug

length, and local pressure gradient. There is also a reported critical

Cain which a complete bypass occurs. This critical value has beenreported for a few channel shapes such as squares and triangles.

Fig. 12. (a) Snapshots of the slug formation process and (b) pressure variation at

the gas and liquid inlets, time points AD correspond to the four snapshots in (a)(Yu et al., 2007).



13/17

As can be seen inTable 3most of the recent attention on veloc-

ity field and flow pattern of two-phase flow inside non-circular

microchannels was on square microchannels. Rectangular micro-

channels were of interest by Yu et al. (2007) and Onea et al.

(2009). Liu and Wang (2008)is one of the rare numerical studies

on flow pattern inside triangular microchannels. Therefore rectan-

gular and triangular geometries need more numerical focus.

Worner et al. (2007)reported that the velocity profile in the li-

quid slug has the same parabolic form for different lengths of com-

putational domain in square microchannels.

Taha and Cui (2006a)showed numerically that the behavior of

dimensionless bubble velocity of Taylor flow in square microchan-

nels is similar to circular microtubes, except for low Cawhere it in-

creases withCa. They also showed that the transition to complete

bypass occurs aroundCa= 0.4. According to their results, two sep-

arate circulation regions exist in front of and behind the bubble at

low Ca. AsCa increases, similar to the circular microtubes, the cen-

ter of circulations moves away from walls and shifts toward the

symmetry line. At highCa, recirculation regions vanish and a com-

plete bypass occurs. They showed that the liquid flow field is not

axisymmetric at low Ca by presenting two different positions of

the vortex eye from the side and diagonal views.

Ghidersa et al. (2004)reported that for the cross section with

the smallest film thickness, in the liquid film surrounding the bub-

ble a back flow region exists. They argued this flow corresponds to

a local high positive pressure gradient due to the rapid change in

film thickness.

Liu and Wang (2008) like other researchers (Taha and Cui,

2006b; Raimondi et al., 2008) reported that with increases in Ca,

the distribution of the vortex centers transferred from walls into

the circle gradually, and become smaller and smaller. They argued

that as the vortex centers shifted toward the centerline, more li-

quid flows towards the bubble and this forms a thicker film. Finally

at critical Ca, at which the bubble velocity is equal to the maximum

liquid velocity, there is a complete bypass and no circulation in li-

quid plug. They obtained a critical Ca of 0.8 and 1.0 in square and

equilateral triangular shapes, respectively.

4.4. Bubble and interface shapes

The main parameters, which are reported to have strong effects

on the bubble shape and interface between the two phases, are Ca

and superficial velocity ratio.

According to the numerical research on bubble shape in square

microchannels, the bubble shape is nonaxisymmetric and flatten

out against the wall for low Ca and axisymmetic, i.e. the bubble

cross section at any axial position remains circular, for high Ca.

Ghidersa et al., 2004; Taha and Cui, 2006b reported that the bubble

shape remains circular even up toCa= 0.043, which they argued is

in agreement with other published studies. However the generally

accepted minimum value of the Cais 0.04. There are several differ-ent reports on transition Ca limit from 0.1 to 0.04 (Taha and Cui,

2006b; Raimondi et al., 2008).

Taha and Cui (2006b)showed that at low Ca, both the front and

rear ends of the bubbles are nearly spherical. When Ca increases,

the convex bubble end inverts gradually to concave. As the Ca in-

creases, the bubble becomes longer and more cylindrical. At higher

Ca numbers, we have cylindrical bubbles.

Liu and Wang (2008) carried out a numerical study on bubble

shape with respect toCa through square and equilateral triangular

microchannels and (similar toTaha and Cui (2006b) and Raimondi

et al. (2008)) argued that because the liquid film thickness in cor-

ners is larger than side walls, the liquid flows more slowly in the

corners. They also reported the transition Calimit of 0.1 for square

microchannels and discussed two potential causes for differencesbetween this limit and the value of 0.04 reported by other

researchers. Firstly, the inertial effect was considered in their study

and second, they simulated a finite long bubble in their work.

Yu et al. (2007) reported that a larger gas to liquid flow ratio

leads to longer gas bubbles in square and rectangular microchan-

nels. Under the same flow rate ratio, a smaller Ca number results

in a longer bubble. According to their results, the geometry of

the mixing section also affects the bubble size and the spacing be-

tween the bubbles.

Reznik and Yarin (2002) simulated numerically the change in

the contact angle of a droplet squeezing between two moving walls

and reported that the free surface deviates from circularity at the

beginning of the squeezing process and after sufficiently strong

squeezing, the contact angle gradually goes up toward the value

ofp. According to their results, the free surface becomes more sim-ilar to the circular shape during the final steps of squeezing.

4.5. Pressure drop

The pressure drop or wall shear stress for the slug flow inside

non-circular channels has not been studied extensively and only

the effects ofCa and flow direction (gravity effects) were of interest

in recent years. Yu et al. (2007) shows similar trend in pressure

across the centerline to the centerline pressure distribution of cir-

cular microtubes, and showed that pressure is distributed almost

constantly along the bubbles with an effective pressure drop inside

the liquid slugs.

Taha and Cui (2006b) investigated numerically wall shear stress

of Taylor flow in square microchannels for two opposite directions

(upward and downward). According to their results, the side wall

shear stress distributions are the same for both flow directions at

low Ca. They also showed that similar to circular capillaries at

lowCa, there are some fluctuations in shear stress profile and that

the central region of the profile (related to bubbles body) has near-

zero value. They argued that the corner wall shear stress for up-

ward flow is higher than the downward flow, because of the differ-

ent liquid film thickness in these two cases.

4.6. Heat transfer

While non-circular microchannels include a variety of geome-

tries like square, rectangle, parallel plates, triangle, and ellipse,

only a few numerical studies were found for moving plugs or drop-

lets between parallel plates. This may be due to the necessity of

using three dimensional computational domains in the cases other

than parallel plates in non-circular microchannels which makes

the numerical studies time-consuming. Table 4 shows the different

numerical works on two-phase heat transfer. As can be seen inTa-

ble 4, significant research gaps exist in the area other than parallel

plates. Non-symmetrical thermal boundary conditions have been

neglected as well. The prominent parameters which affect heat

transfer in non-circular channels have been reported as Pe, sluglength, channel geometry, interface shape (contact angle), and flow

pattern (internal circulations).

Baird (2008) showed that shorter droplets moving between two

parallel plates have higher values ofNu and as the length of the

droplets increases the heat transfer characteristics moves toward

the single phase heat transfer, because the main mechanism of

increasing convective heat transfer i.e. internal circulations in the

droplet has less effect.Baird (2008)also mentioned that important

parameters in heat transfer modeling for moving droplets between

parallel plates are Pe, droplet length, distance between two parallel

plates, and meniscus curvature effects i.e. contact angle. As a result,

the reports ofNu versus position of droplets should include all of

the aforementioned parameters.

Young and Mohseni (2008) reported higher Nu (after initiallyoscillating) for various droplet sizes moving between parallel



14/17

plates compared with continues Graetz flow. They argued this is

because of the circulating internal flows inside the moving drop-

lets. They also showed that as the droplets become shorter, the

Nuincreases because there is relatively more cool fluid inside the

droplet which can be brought into contact with the heated wall

in comparison to the longer droplets. They reported thatNubegins

to decrease (before being constant) when the first circulation com-

pletes and the previously heated fluid is brought back to the heated

walls. They also reported a locally higher value ofNubefore stabil-ization and argued this as follows. In long droplets the dominant

heat transfer process is diffusion from the wall into the liquid in ra-

dial direction, but in short droplets the diffusion in wall-tangent

direction has the main role. The highest peak in Nu occurs when

these two diffusion times become near equal in magnitude.

Oprins et al. (2008) studied numerically the internal flow pat-

terns and heat transfer inside electrostatic actuated droplets mov-

ing between two parallel plates. They carried out steady and time-

dependent simulations, assumed laminar flow and constant fluid

properties. They also proposed a lumped model by dividing the

droplet into four cells for steady state case, and then expanded it

in order to make it capable of predicting un

Talimi 2012 International Journal of Multiphase Flow

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