Modeling of Gas Liquid Taylor Flow in Capillaries by Using a Two

Article pubs.acs.org/IECR

Modeling of Gas Liquid Taylor Flow in Capillaries by Using a Two Fluid Model Bin Cao†,‡ and Lingai Luo§,* †

Dalian National Laboratory for Clean Energy, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People's Republic of China ‡ School of Chemical Engineering, Northwest University, 229 Taibai Road, Xi’an, 710069, People's Republic of China § Laboratoire Optimisation de la Conception et Ingénierie de l’Environnement (LOCIE), CNRS UMR 5271, Université de Savoie, Campus Scientifique, Savoie Technolac, 73376 Le Bourget-Du-Lac Cedex, France ABSTRACT: This work proposes a two fluid model for predicting gas liquid Taylor flow in capillaries. The model is based on the hydrodynamics characteristics of Taylor bubbles and liquid slug. By considering a unit cell consisting of a bubble and a liquid slug as control volume, conservation of momentum equations for gas and liquid film is combined and solved with a set of empirical closure equations. We also propose a new expression for the interfacial friction factor, which takes the influences of surface tension, the liquid velocity and viscosity and the gas holdup into account. By combining this new expression with the two fluid model, we can calculate the hydrodynamic characteristics of gas−liquid Taylor flow in capillaries for different channels diameters and working fluids. The model is validated by experimental results reported in existing literature. Very good agreement can be observed, demonstrating its potential application in predicting the phenomena of Taylor flow in capillaries.

1. INTRODUCTION Gas liquid two phase flow in capillaries is a very common occurrence in microreaction technology, because a large number of mass transfer and reaction processes involve both the gas and the liquid phase.1,2 When gas and liquid simultaneously exist in a capillary, it will take on a variety of configurations or patterns related to different spatial distributions of multiphase interface.3−6 The two-phase flow pattern is contingent on many factors, such as the gas and liquid velocity, the physical chemical properties of working fluids, and the geometry of the capillaries.7−11 Previous studies5 have demonstrated that gas liquid flow in capillary can be classified into five main patterns: bubbly flow, Taylor flow (including the unsteady Taylor flow), slug-annular flow, annular flow, and churn flow. In all these patterns, the Taylor flow is one of the most important flow regimes occurring in gas−liquid capillaries, because this pattern may appear under a wide range of gas and liquid velocity conditions.12 As a result, to understand the mechanism of Taylor flow in capillaries is in fact very important for the design and application of efficient multiphase micro reactors. The approaches for predicting the Taylor flow include multiphase Computational Fluid Dynamic (CFD) method12−14 and empirical or approxiamtion correlations method.4,5,12 The multiphase CFD method is more detailed, but it is more complicated and involves a large amount of computations. We still do not have a simple and effective model to calculate and analysis the pressure drop and bubble velocity of gas liquid Taylor flow in capillaries. In 1935, Fairbrother et al.15 first proposed a experimental empirical correlation to express the relation between the bubble velocity and the capillary number. Furthermore, Bretherton16 used lubrication theory to analyze long bubbles moving steadily in a circular capillary. On the basis of the asymptotic approximation in the limit of low capillary number, he proposed two approximation equations to predict © 2012 American Chemical Society

the bubble velocity and pressure drop with the capillary number. Ratulowski et al.17 and Kolb et al.18 used lubrication analysis to investigate the transport of long bubbles in capillaries and tubes of square cross section, and they proposed some new expressions to improve the results of Bretherton. In spite of its simplicity, the most obvious deficiency of empirical or approximation correlation methods is that it concentrates only on the motion of the bubble, without considering the whole movement of the bubble and the liquid slug, and thus cannot be used to predict the bubble velocity and pressure drop simultaneously or under the same frame. The two fluid model method is a very effective tool for the analysis and simulation of gas liquid annular and stratified flow in pipes. This method considers gas and liquid as two penetrating mutually continuous media, the movement of gas and liquid phases can be described simultaneously by two different control equations. In particular, the momentum exchange between the two phases is controlled by the local shear stress at the gas−liquid interface. As a result, the interfacial friction factor is a key parameter for the determination of the gas liquid two phase flow behaviors such as the relative translational velocity of bubbles and the two phase pressure drop. This method has been successfully applied in predicting the steady-state and transient gas−liquid two phase slug flow and stratified flow in pipes and conduits.19−21 As far as we know, the two fluid model has not been applied in the study of the gas liquid Taylor flow in capillaries. In this work, we try to propose a semiempirical model based on two fluid model for simulating the gas liquid Taylor flows in Received: Revised: Accepted: Published: 13054

March 27, 2012 July 31, 2012 August 29, 2012 August 29, 2012 dx.doi.org/10.1021/ie300817f | Ind. Eng. Chem. Res. 2012, 51, 13054−13061

Industrial & Engineering Chemistry Research

Article

transfer. Through combining this new expression with the two fluid model, we are able to calculate the hydrodynamic characteristics of the gas−liquid Taylor flow in capillaries for different channels diameters and working fluids. The model will be validated by the experimental results of Liu et al.,4 and the predicted total pressure drop and bubble velocity will also be compared with the relevant experimental data.

circular and square capillaries. The existing interfacial friction factor correlations used in the two fluid model are only applicable under high Reynolds number conditions and for large diameter pipes. The available correlations for gas liquid interfacial friction factor only consider the influence of two phase inertial force on the interfacial friction behaviors. However, previous research22−24 has shown that, the influence of surface tension on interfacial stress boundary condition cannot be ignored. Therefore, a new correlation for calculation of the interfacial friction factor should be proposed, this formula will integrate the inertial force and surface tension on the interfacial momentum

2. MATHEMATICAL FORMULATION 2.1. Two Fluid Model for Taylor Flow. As shown in Figure 1, a Taylor unit cell is used as the control volume, which consists of a gas bubble, a liquid slug, and a liquid film between the bubble and the wall. Mass flow conservation for gas and liquid systems: ∂ρg αg ∂t ∂ρl αl ∂t

∂ρg αg Ug

+

∂z ∂ρl αlUl

+

∂z

=0 =0

(1)

Momentum balance for gas and liquid systems, ∂ρg αg Ug ∂t

+

= −αg ∂ρl αlUl ∂t

∂ρg αg Ug2 ∂z

∂Pg ∂z

+

− ρg αg g sin θ −

∂ρl αlUl2 ∂z

= −αl

τgSg A

−

τiSi A

(2)

∂Pl τS τS − ρl αlg sin θ − l l + i i ∂z A A

Figure 1. Control volume using in calculation.

(3)

Table 1. Some Correlations for Gas Liquid Interfacial Friction Factor proposer

correlation

Asali J. C. et al.27

suitable conditions vertical gas liquid annular flow

for upflows fi /fs = 1 + C1(mg+ − 4) mg+ = 0.34 Re0.6 LF

1/2 vL ⎛ ρL τi ⎞ ⎜⎜ ⎟⎟ vG ⎝ ρG τC ⎠

for gas velocities greater than25m/s τ C1 = 0.045 and i ≈ 1. τC for more detail, please refer to the reference. Andritsos N. et.al.28

gas liquid stratified flow in a horizontal pipe

⎛ ρ ⎞0.5 g0 fi = 16/Reg for UGS ≤ ⎜⎜ ⎟⎟ ⎝ ρg ⎠ fi =

0.5 ⎛ ⎤⎞ ⎡ ⎛ h ⎞0.5⎢ UGS ⎛ ρg 0 ⎞ 16 ⎜ ⎜ ⎟ − 1⎥⎟ 1 15 + ⎟ ⎜ ⎥⎟⎟ Reg ⎜⎜ ⎝ dh ⎠ ⎢⎢ 5 ⎜⎝ ρg ⎟⎠ ⎥⎦⎠ ⎣ ⎝

Reg ≤ 2, 000 h is the height of the film ρg 0 is the gas density at atmosphere pressure Kowalski J. E.

29

fi = 7.5 × 10−5(1 − αg )−0.25 Re−g 0.3Rel0.83

gas liquid stratified flow in a c horizontal pipe

22, 600 ≤ Reg ≤ 430, 600 30

Newton C. H.

0.6 fi = 8.9 × 10−7hL−0.5Re0.41 g Rel dh = 50mm

gas liquid stratified flow in a horizontal pipe

fi = 8.6 × 10−7hL−0.23Re−g 0.19Re1.2 l dh = 80mm 13055

dx.doi.org/10.1021/ie300817f | Ind. Eng. Chem. Res. 2012, 51, 13054−13061


Article

Figure 2. (a) Variation of the experimental and predicted gas holdup εG with different superficial gas and liquid velocities for 2 mm circular air capillary, air−water systems; □, ULS = 0.664 m/s; ×, ULS = 0.492 m/s; ○, ULS = 0.309 m/s; and Δ, ULS = 0.138 m/s. (b) Variation of the experimental and predicted total pressure drop with different superficial gas and liquid velocities for 2 mm circular air capillary, air−water systems: □, ULS = 0.664 m/s; ×, ULS = 0.492 m/s; ○, ULS = 0.309 m/s; and Δ, ULS = 0.138 m/s.

Figure 3. (a) Variation of the experimental and predicted gas holdup εG with different superficial gas and liquid velocities for 3.02 mm circular air capillary, air−water systems: □, ULS = 0.455 m/s; ×, ULS = 0.363 m/s; ○, ULS = 0.268 m/s; and Δ, ULS = 0.159 m/s. (b) Variation of the experimental and predicted total pressure gradient with different superficial gas and liquid velocities for 3.02 mm circular air capillary, air−water systems: □, ULS = 0.455 m/s; ×, ULS = 0.363 m/s; ○, ULS = 0.268 m/s; and Δ, ULS = 0.159 m/s.

where αg and αl is the gas and liquid phase fraction at the cross section. Ug and Ul is the gas and liquid phase real velocity, Pg and Pl is the gas and liquid phase pressure, τg andτl represent the gas and liquid wall shear stress respectively and τi represents the shear stress at the interface. Expressions for the geometrical quantities are calculated by the following: αg = Ag /A αl = Al /A

(4)

A = πdh2/4Ag = πdb2/4 A f = A − Ag As = A

(5)

Sg = 0; Sf = πdh ; Si = πdb ; Ss = πdh

(6)

mg = UgAg ρg

Lg Lc

(8)

Where Ls , Lg, and Lc is the length of the liquid slug, gas bubble and unit cell respectively. And mg and ml represent the gas phase and liquid phase mass flow rate. In order to effectively solve above momentum balance differential equations of gas and liquid (2) ∼ (3), some assumptions should be made: - Uniform axial direction velocity profiles in a bubble, film, and liquid slug; - Steady state two phase flow.

where A is the cross sectional area of the pipe, Af is the cross sectional area of the liquid film, Ag and Al is the cross sectional area of the gas and the liquid phase, As is the cross sectional area of the liquid slug. Sf is the wetted perimeters for liquid film, Sg and Sl is the wetted perimeters for gas and liquid. Si is the Taylor bubble perimeters. Ss is the wetted perimeters for liquid slug. Integrating the differential eqs 1 at a fixed cross section over the time of passage of a Taylor unit yields following algebraic equations: Lg Ls ml = UA + Uf A f ρl s ρl Lc Lc (7)

By using above assumptions and simplifications, we can combine conservation of momentum equations of gas and liquid systems with following equations. For the flow in liquid slug region: ∂Pl τS + ρl g sin θ + l l = 0 ∂z A 13056

(9)



Article

Figure 5. (a) Comparison of experimental bubble rise velocity with predicted values for the circular capillary air−water system. □, dh = 0.91 mm; ◊, dh = 2.00 mm; and Δ, dh = 3.02 mm. (b) Comparison of experimental total pressure drop with predicted values for the circular capillary air−water system. □, dh = 0.91 mm; ◊, dh = 2.00 mm; and Δ, dh = 3.02 mm.

Figure 4. (a) Variation of the experimental and predicted gas holdup εG with different superficial gas and liquid velocities for 2.00 mm circular capillary, air−ethanol systems: ○, ULS = 0.537 m/s; Δ, ULS = 0.343 m/s; and ■, ULS = 0.124 m/s m. (b) Variation of the experimental and predicted total pressure gradient with different superficial gas and liquid velocities for 2.00 mm circular capillary, air−ethanol systems: ○, ULS = 0.537 m/s; Δ, ULS = 0.343 m/s; and ■, ULS = 0.124 m/s m.

where τ*i = τi + τsti is defined as total interfacial shear stress, which represents the total influence of two phase fluid flow and the interphase pressure jump on the inter phase momentum transfer. 2.2. Calculation of Shear Stresses. The shear stresses in the combined momentum equations eq 12 are determined in terms of the real gas and liquid velocities as follows (see Figure 1 for the definition of the positive direction for the fluid flow):

For the flow in gas bubble and liquid film region: τf

Sf

− τg

Af

−

⎛1 1 ⎞⎟ − τiSi⎜⎜ + + g (ρl − ρg )sin θ Ag Ag ⎟⎠ ⎝ Af Sg

∂(Pg − Pf ) ∂z

=0

(10)

Here, we define an additional interfacial phase shear stress τsti , given by the following: st

τi =

∂(Pg − Pf ) ∂z

⎛1 1 ⎞⎟ /Si⎜⎜ + Ag ⎟⎠ ⎝ Af

⎛1 1 ⎞⎟ − τi*Si⎜⎜ + + g (ρl − ρg )sin θ = 0 Af Ag ⎟⎠ ⎝ Af

(13)

τs = fs (ρl |Us|Us/2)

(14)

τi* = f i* (ρg |Ug − Uf |(Ug − Uf )/2)

(15)

fg, f f, f i*

where are the friction factors between the gas and the wall, the liquid film and the wall, and the gas liquid interface, respectively. The liquid film/wall friction factors for circular and square capillary f f, are evaluated from the following:19,25

(11)

where Pf is the liquid film pressure. Above equation represents the influence of the pressure jump across the gas liquid interface on the inter phase momentum transfer. According to the dynamic free surface boundary condition,16,22−24 this pressure jump is a function of the surface tension, local interface curvature, and the characteristics of gas liquid buoyancy flow. Substituting eq 11 into eq 10, we get, τf

τf = f f (ρl |Uf |Uf /2)

f f = 16/(df Uf ρl /μl ) (for a circular capillary under laminar flow)

(16)

f f = 14.25/(df Uf ρl /μl )

Sf

(for a square capillary under laminar flow)

(12) 13057

(17)



Article

Figure 7. (a) Comparison of experimental bubble rise velocity with predicted values for the 3.02 mm circular capillary air−oil system. (b) Comparison of experimental total pressure gradient with predicted values for the 3.02 mm circular capillary air−oil system.

Figure 6. (a) Comparison of experimental bubble rise velocity with predicted values for the circular capillary air−ethanol system. □, dh = 0.91 mm; ◊, dh = 2.00 mm; and Δ, dh = 3.02 mm. (b) Comparison of experimental total pressure gradient with predicted values for the circular capillary air−ethanol system. □, dh = 0.91 mm; ◊, dh = 2.00 mm; and Δ, dh = 3.02 mm.

proposed many correlations to calculate the interfacial friction factor for gas liquid in pipes with large diameter (see Table 1). However, these correlations are not suitable for predicting the Taylor flow in capillaries, since they have not taken the influences of surface tension into account and only suitable for high Reynold number conditions. In this study, we propose a new correlation as follows:

where the hydraulic diameter of the liquid film df, is given by the following:

df = 4A f /Sf

(18)

The liquid slug/wall friction factor fs for circular and square capillary can also be calculated by eq 16-18 by replacing the subscript f with s.19 Kreutzer et al.26 have proposed two expressions for estimating the diameter of the bubble in circular and square capillaries, given as follows: db = 0.64 + 0.36exp(− 2.13Ca 0.52) dh

f i* =

dbUgρg /μg

(21)

where, εg = UGS /Ug Ca = μl (UGS + ULS)/σ Rel = dhULSρl /μl (22)

(for a circular capillary)

and in this work we chose a = 0.1364; b = −0.0424; c = 0.24; d = 0.7834. These values are obtained by optimal fit of data points of experimental results of Liu et al.4 Note that circular shape cross-section of Taylor bubble is experimentally observed for both circular and square capillaries. The mechanism for the friction at gas/liquid interface should be identical, not directly related to the cross-sectional shape of capillaries. Therefore, a single correlation eq 21 is proposed for calculating the interfacial frictional factor for both circular and square capillaries. 2.3. Total Pressure Gradient Across a Taylor Unit Cell. We use the cross section average pressure to characterize the local two-phase pressure, which is given by the following: P = αg Pg + αlPl (23)

(19)

db = 0.7 + 0.5exp(− 2.25Ca 0.445) dh

aCa b(1 − εg )/εgc Reld(μl /μg )

(for a square capillary) (20)

where Ca is the capillary number. The interfacial friction factor is a key parameter in modeling Taylor flow in capillaries, because of its direct expression of various flow and the physical parameters of the effect of momentum transfer. It is very difficult to deduce the expression of the interfacial friction factor directly from the theory of the gas liquid two phase fluid dynamic, and it can only be obtained through the semiemprical correlating of the experimental results. Researchers27−30 have 13058



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Figure 9. (a) Comparison of experimental bubble rise velocity with predicted values for the square capillary. □, dh = 2.89 mm, air−water; ◊, dh = 0.99 mm, air−ethanol; Δ, dh = 2.89 mm, air−ethanol. (b) Comparison of experimental total pressure gradient with predicted values for the square capillary. □, dh = 2.89 mm, air−water; ◊, dh = 0.99 mm; air−ethanol; Δ, dh = 2.89 mm, air−ethanol.

Figure 8. (a) Variation of the experimental and predicted gas holdup εG with different superficial gas and liquid velocities for 2.89 mm square capillary, air−ethanol systems: × ULS = 0.346 m/s; ○, ULS = 0.247 m/s. (b) Variation of the experimental and predicted total pressure gradient with different superficial gas and liquid velocities for 2.89 mm square capillary, air−ethanol systems: ×, ULS = 0.346 m/s; ○, ULS = 0.247 m/s.

system in a circular capillary with a 2 mm diameter. It increases monotonically with the increasing UGS. Figure 2b shows the total pressure gradient as a function of UGS, at different liquid velocities (ULS). It can be observed that at low ULS, the total pressure gradient decreases with the increasing UGS. When liquid flow increases, this tendency is reduced gradually. In fact, when gas velocity increases, the gas hold up increases too. This results in the decrease of average density of the slug unit. Then the pressure gradient is significantly reduced, especially at low liquid flow. The similar tendency is observed at larger diameter (Figure 3a,b) and for air-ethanol system (Figure 4a,b). In all of these figures, it can be observed that our simulation results are in good agreement with the experimental data. For an air−water system in circular capillary with different diameters, less than 20% deviation is obtained for bubble velocity(Figure 5a), the deviation for total pressure is largely within ±30% (Figure 5b). For an air−ethanol system in circular capillary with different diameters, the deviation for both pressure drop and bubble velocity is roughly within ±20%, as shown in Figure 6a,b. Figure 7a,b indicates the comparison for air−oil system. Good agreement between model predictions and the measurements can be observed, except for several data points corresponding to relatively low gas flow rates. 3.2. Square Capillaries. Figure 8a,b shows the comparison results of air−ethanol two phase flow in square capillaries. It is seen that the model gives good predictions for both the total

Using a global force balance along a unit cell, the total pressure gradient across this control volume can be determined by the following: τf Sf + τgSg Lg ⎛ dP ⎞ τS L ΔP ⎜ − ⎟ = = ρc g sin θ + s s s + ⎝ dz ⎠T Lc A Lc A Lc

(24)

where ρc is the average density of the Taylor unit, given as follows: ρc = εg ρg + (1 − εg )ρl

(25)

and in this study, tubes were placed vertically, which means θ = π/2.

3. RESULTS AND DISCUSSION The model validation is carried out by comparing the total pressure gradient, gas holdup, and bubble rising velocity with relevant raw experimental data provided by the paper of Liu et al.4 The detail of experimental conditions and working fluid properties can also be found in their paper. Note that in all of the figures presented in following subsections, the lines represent our simulation results, and the dotes are experimental data obtained from Liu et al.4 3.1. Circular Capillaries. Figure 2a shows the gas hold-up as a function of superficial gas velocity (UGS) for the air−water 13059



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fg f i, f *i fs g Ls, Lg Lc

pressure gradient and the gas hold-up. Figure 9a,b shows the comparison results of two phase flow in square capillaries for different working fluids. Again, the overall agreement is good between the model predictions and Liu’s measurements. It is worth pointing out that for the reason of the experimental detection, the calculation results are only compared with the overall quantities such as pressure drop and bubble velocity. The investigation of more detailed parameters such as interfacial friction may be beyond the applicable scope of the model. For that case, micro scale flow field measurement and CFD simulations are necessary, and these are our ongoing work. In addition, the model is based on the fitting formula derived from the fitting of Liu’s experimental data. Therefore, we can not say that this formula is universal. The only thing we can claim is that this model is suitable for predicting the hydrodynamics of gas−liquid Taylor flow in capillaries under the experimental conditions used in Liu’s work. We still need more extensive experimental data and results to validate the universality of our model.

mg, ml P ΔP Pf Pg , Pl Rel Sf Sg , Sl, Si Ss t Us Uf Ug,Ul UGS, ULS z

4. CONCLUSIONS A new correlation for calculation of the interfacial friction factor has been proposed in this work, this formula integrated the influence of inertial force and surface tension on the interfacial momentum transfer. Combining this new expression with the two fluid models, we are able to predict the hydrodynamic characteristics of the gas−liquid Taylor flow in capillaries for different channels diameters and working fluids. The mechanistic model has been validated with experimental measurements of total pressure gradient and gas hold-up. For different channel diameters and working fluids, our numerical results obtained are in good agreement with the experimental data. It is shown that this model can be applied to analyze and predict the Taylor flow phenomenon in capillaries.

■

Greek Symbols

αg,αl θ τ ρg,ρl

■

AUTHOR INFORMATI ON

*Laboratoire de Thermocinétique, UMR CNRS 6607, Ecole PolytechniqueUniversity of Nantes, La Chantrerie, Rue Christian Pauc, BP 50609, 44306 Nantes Cedex, France. Phone: +33 4 79 75 81 93. Fax: +33 4 79 75 81 44. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is financially supported by French ANR within the project MIGALI (ANR-09-BLAN-0381-1) and by Nature Science Foundation of Shaanxi Province (No.2007B14), and Natural Science Foundation of Education Committee of Shaanxi Province (No. 07JK380). A Af Ag, As Ca db df dh ff

= = = =

gas or liquid fraction at the cross section inclination angle of the tube shear stresses kg/m/s2 density for gas and liquid phase kg/m3

REFERENCES

(1) Hessel, V.; Angeli, P.; Gavriilidis, A.; Lowe, H. Gas-liquid and gasliquid-solid microstructured reactors: Contacting principles and applications. Ind. Eng. Chem. Res. 2005, 44, 9750. (2) Jahnisch, K.; Baerns, M.; Hessel, V.; Ehrfeld, W.; Haverkamp, V.; Lowe, H.; Wille, C.; Guber, A. Direct fluorination of toluene using elemental fluorine in gas/liquid microreactors. J. Fluorine Chem. 2000, 105, 117. (3) Laborie, S.; Cabassud, C.; Durand-Bourlier, L.; Laine, J. M. Characterisation of gas-liquid two-phase flow inside capillaries. Chem. Eng. Sci. 1999, 54, 5723. (4) Liu, H.; Vandu, C. O.; Krishna, R. Hydrodynamics of taylor flow in vertical capillaries: Flow regimes, bubble rise velocity, liquid slug length, and pressure drop. Ind. Eng. Chem. Res. 2005, 44, 4884. (5) Liu, D. S.; Wang, S. D. Flow pattern and pressure drop of upward two phase flow in vertical capillaries. Ind. Eng. Chem. Res. 2008, 47, 243. (6) Yue, J.; Luo, L.; Gonthier, Y.; Chen, G.; Yuan, Q. An experimental investigation of gas-liquid two-phase flow in single microchannel contactors. Chem. Eng. Sci. 2008, 63, 4189. (7) Suo, M.; Griffith, P. Two Phase Flow in Capillary Tubes; Massachusetts Institute of Technology: Cambridge, MA, 1963. (8) Suo, M.; Griffith, P. Two-phase flow in capillary tubes. J. Basic Eng. 1964, 86, 576. (9) Brauner, N.; Maron, D. M. Identification of the range of small diameters conduits, regarding 2-phase flow pattern transitions. Int. Commun. Heat Mass Trans. 1992, 19, 29. (10) Kew, P.; Cornwell, K. Correlations for prediction of boiling heat transfer in small diameter channels. Appl. Thermal Eng. 1997, 17, 705. (11) Shao, N.; Gavriilidis, A.; Angeli, P. Flow regimes for adiabatic gas-liquid flow in microchannels. Chem. Eng. Sci. 2009, 64, 2749. (12) Kreutzer, M. T.; Kapteijn, F.; Moulijin, J. A.; Kleijn, C. R.; Heiszwolf, J. J. Inertial and interfacial effects on pressure drop of Taylor flow in capillaries. AIChE J. 2005, 51, 2428. (13) Akbar, M. K.; Ghiaasiaan, S. M. Simulation of Taylor flow in capillaries based on the volume-of-fluid technique. Ind. Eng. Chem. Res. 2006, 45, 5396.

Corresponding Author

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= friction factors between the gas and the wall = friction factors between the gas liquid interface = friction factors between the liquid slug and the wall = acceleration of gravity (m s−2) = length of the liquid slug, gas bubble length and unit celll = gas phase and liquid phase mass flow rate (kg/s) = cross section average pressure (Pa) = total pressure drop (Pa) = liquid film pressure (Pa) = gas phase and liquid phase pressure (Pa) = Reynolds number for liquid phase (dhULSρl/μl) = wetted perimeters for liquid film (m) = wetted perimeters for gas and liquid (m) = Taylor bubble perimeters(m) = wetted perimeters for liquid slug (m) = time (s) = liquid slug velocity (m/s) = liquid film velocity (m/s) = gas phase and liquid phase real velocity (m/s) = gas phase and liquid phase superficial velocity (m/s) = axial coordinate (m)

NOMENCLATURE = cross sectional area of the pipe (m2) = cross sectional area of the liquid film (m2) Al = cross sectional area of the gas and the liquid (m2) = cross sectional area of the liquid slug (m2) = capillary number (μl (UGS+ULS)/ σ) = hydraulic diameter of the Taylor bubble (m) = hydraulic diameter of the liquid film (m) = hydraulic diameter of the pipe (m) = friction factors between the liquid film and the wall 13060



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(14) Feng, J. Q. A long gas bubble moving in a tube with flowing liquid. Int. J. Multiphase Flow 2009, 35, 738. (15) Fairbrother, F.; Stubbs, A. E. Studies in electro-endosmosis. Part VI. The ‘bubble-tube’ method of measurement. J. Chem. Soc. 1935, 1, 527. (16) Bretherton, F. P. The motion of long bubbles in tubes. J. Fluid Mech. 1961, 10, 166. (17) Ratulowski, J.; Chang, H. C. Transport of gas bubbles in capillaries. Phys. Fluids A. 1989, 10, 642. (18) Kolb, W. B.; Cerro, R. L. The motion of long bubbles in tubes of square cross section. Phys. Fluids A. 1993, 5, 1549. (19) Taitel, Y.; Barnea, D. Two phase slug flow. Adv. Heat Trans. 1990, 20, 83. (20) Zhang, H. Q.; Wang., Q.; Sarica, C.; Brill, J. P. Unified model for gas liquid pipe flow via slug dynamicsPart 1: Model development. Trans. ASME 2003, 125, 266. (21) Johnson, G. W.; Bertelsen, A. F.; Nossen, J. A mechanistic model for roll waves for two-phase pipe flow. AIChE J. 2009, 55, 2788. (22) Janicot, A.; Dukler, A. E. A model for gas liquid slug flow at reduced gravity conditions. AIChE J. 1993, 39, 1101. (23) Taitel, Y.; Witte, L. The role of surface tension in microgravity slug flow. Chem. Eng. Sci. 1996, 51, 695. (24) Pozrikidis, C. Interfacial dynamics for stokes flow. J. Comput. Phys. 2001, 169, 250. (25) Shah, R. K.; London, A. L. Laminar Flow Forced Convection in Ducts, Supplement 1 to Advances in Heat Transfer; Academic: New York, 1978. (26) Kreutzer, M. T.; Du, P.; Heiszwolf, J. J.; Kapteijn, F.; Moulijn, J. A. Mass transfer characteristics of three-phase monolith reactors. Chem. Eng. Sci. 2001, 56, 6015. (27) Asali, J. C.; Hanratty, T. J.; Andreussi, P. Interfacial drag and film height for vertical annular flow. AIChE J. 1985, 31, 895. (28) Andritsos, N.; Hanratty, T. J. Influence of interfacial waves in stratified gas liquid flow. AIChE J. 1987, 33, 444. (29) Kowalshi, J. E. Wall and interfacial shear stress in stratified flow in a horizontal pipe. AIChE J. 1987, 33, 274. (30) Newton, C. H.; Behnia, M. On the use of momentum balance for the deduction of shear stress in horizontal gas liquid pipe flow. Int. J. Multiphase Flow 1998, 24, 1407.

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Modeling of Gas Liquid Taylor Flow in Capillaries by Using a Two

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