# Wide Range Simulation Study of Taylor Bubbles in Circular Milli and Microchannels

^{*}

## Abstract

**:**

_{B}) (0.01–2) and Reynolds numbers (Re

_{B}) (0.01–700). The shape and velocity of the bubbles are, together with the flow patterns in the flowing liquid, analyzed and compared with numerical and experimental correlations available in the literature. For low values of Ca

_{B}, the streamlines (moving reference frame (MRF)) in the liquid slug show semi-infinite recirculations occupying a large portion of the cross-section of the channel. The mean velocity of the fluid moving inside the external envelope of the semi-infinite streamlines is equal to the bubble velocity. For high values of Ca

_{B}, there are no recirculations and the bubble is moving faster or at least at the velocity of the liquid in the center of the tube; this flow pattern is often called bypass flow. The results also indicate that the liquid film surrounding the bubbles is for low Ca

_{B}and Re

_{B}numbers almost stagnant, and its thickness accurately estimated with existing correlations. The stagnant film hypothesis developed provides an accurate approach to estimate the velocity of the bubble, in particular for low values of Ca

_{B}. The asymptotic behavior of the studied parameters enables the extrapolation of data for Ca

_{B}lower than 0.01. In addition to the simulations of isolated bubbles, simulations with two consecutive bubbles were also carried out; coalescence was only observed in very specific conditions. The results obtained in this study are directly applicable to co-current slug flow in milli- and microchannels for 0.1 < Re

_{B}< 1000 and 0.02 < Ca

_{B}< 2.

## 1. Introduction

^{2}by Suo and Griffith [8] and 3.37 by Bretherton [9], estimated differently. In channels with dimensions until a few hundred microns, the Eötvös number is far below both of these limits, regardless the properties of the fluids. Therefore, the simulations carried out in the present work neglect the gravity effect.

_{B}is the velocity of the bubble and μ

_{L}the viscosity of the liquid.

_{B}and Ca

_{B}:

## 2. State of the Art

#### 2.1. Liquid Film around the Taylor Bubble

_{B}was proposed:

_{B}, Re

_{B}and We

_{B}to predict liquid film thickness within ±15% of accuracy:

#### 2.2. Taylor Bubble Velocity

_{TP}:

_{TP}is the two-phase velocity, defined as the sum of the gas and liquid superficial velocities.

_{B}) can be one of three different functions, with increasing accuracy but also with increasing complexity, presented below:

_{B}, A

_{film}and A

_{MC}are the cross section areas of the bubble, film and microchannel, respectively, and V

_{film}is the average velocity in the film. If V

_{film}is zero, then the equation is simplified:

#### 2.3. Flow in Liquid Slugs

_{TP}and is independent of the superficial velocity ratio V

_{G}/V

_{L}for the studied conditions.

#### 2.4. CFD Model

_{i}to track the position of the interface. This variable is 1 at any point exclusively occupied by fluid i and 0 if i is not present. In the interface between both fluids, the variable assumes a value between 1 and 0. Therefore,

## 3. Results and Discussion

_{B}or Ca

_{L}) and the Reynolds number (Re

_{B}). Therefore, a series of simulations were carried out in the range of 0.01–2 for Ca

_{B}and 0.01–700 for Re

_{B}. The limits of the studied Ca

_{B}/Re

_{B}map were imposed by several restrictive factors. Below Ca

_{B}= 0.01, spurious currents affect significantly the numerical solution while Ca

_{B}> 2 is unrealistic for flows in milli- or microchannels. In addition, for Re

_{B}higher than 700, the flow becomes 3D.

_{B}versus Ca

_{B}plot in Figure 2.

_{B}or Re

_{B}group they can have slightly different values. This is due to the CFD methodology described above, which involved iterating the bubble velocity. However, as a matter of convenience, the results presented in the next sections will neglect these differences.

^{5}Pa in 100 mm of tube length, while for the micro tubes was less than 4 × 10

^{5}Pa in 1 mm of tube length.

_{B}(<0.01) was not covered due to numerical shortcomings already mentioned (spurious currents). However, as shown later, the main features of the hydrodynamic parameters in this region are easily extrapolated due to their asymptotic behavior.

#### 3.1. Bubble Shape

_{B}causes the front of the bubble to become slenderer for the whole range of Re

_{B}studied. The rear of the bubble tends to become more concave for higher Ca

_{B}, being this effect more accentuated for high Re

_{B}. Additionally, for high Re

_{B}, appear some stationary wavy patterns near the back of the bubble. Figure 8 shows the contours of the bubble for each Re

_{B}studied.

_{B}< 0.3, the streamlines (MRF) in the liquid slug show semi-infinite recirculations, occupying a larger part of the cross-section of the channel. The extent of the cross-section occupied increases as Ca

_{B}decreases. The mean velocity of the fluid moving inside the external envelope of these streamlines is equal to the bubble velocity. The fluid placed ahead the bubble, between the external envelope and the tube wall, moves at a lower velocity than the bubble and will be surpassed by the bubble. As Ca

_{B}decreases, the thickness of the film also decreases, and according to the stagnant film hypothesis (Equation (14)) the bubble velocity also decreases. The center of the recirculation vortex moves away from the center of the channel into the wall direction. Previous studies found in literature report similar recirculations in MRF [27], with many of them focused on the flow pattern in liquid slugs between two bubbles [26,28,33]. This topic will be developed later in the text. For Ca

_{B}= 0.8, there are no recirculations and the bubble is moving faster or at the velocity of the liquid in the center of the tube; this flow pattern is often called bypass flow.

_{B}= 0.8 and Re

_{B}= 100 appears, at the rear of the bubble, a recirculation region similar to that observed in vertical macro studies, significant gravity and inertial effects, in laminar regime, the so called wake region where the fluid circulates in a toroidal vortex [34]. The appearance of this region is a consequence of the expansion of the liquid film at the rear of the bubble.

#### 3.2. Liquid Film

_{B}and Ca

_{B}. The close symbols refer to the simulated cases and the solid lines represent the correlation of Han and Shikazono [17], which predicts the liquid film thickness with ±15% accuracy. As previously referred the film thickness decreases for decreasing Ca

_{B}. According to the figure, for values of Ca

_{B}lower than 0.2 the liquid film thickness is independent of Re

_{B}. For values of Ca

_{B}lower than 0.01, region not covered in the simulations, the liquid film thickness tends asymptotically to zero, i.e., the bubble tends to flow occupying all the cross-section of the tube like a two-phase plug flow.

_{B}, where it is harder to simulate the liquid film; for the highest bubble velocity (Ca

_{B}= 0.110; Re

_{B}= 772.7) where the error is about 23%. This large deviation is probably because under those conditions the liquid film is not yet fully stabilized for that bubble length.

_{B}= 10 it is almost independent of Reynolds number. For Re

_{B}= 100 there is a notable increase of this length for all the Capillary numbers.

_{B}= 0.01 are not presented in Figure 10 since these conditions were simulated with the coupled VOF-level-set method with Heaviside correction to reduce spurious currents. While the solutions obtained with this method are still in agreement with the presented correlations for bubble velocity and liquid film thickness, the method causes the nose and back of the bubbles to have a higher radius of curvature than expected, (this effect is particularly evident in Figure 8). Consequently, the parameter d* is abnormally high in those cases.

_{B}the values of the development length tend to be very small since the liquid film tends to disappear. Further work is necessary to have a better understanding of the behavior of this parameter for Re

_{B}higher than 10

#### 3.3. Bubble Velocity

_{B}, up to a maximum around 12% for the lowest Re

_{B}. This is mainly because this correlation is based on experiments performed with a range of higher Reynolds numbers than the ones used in this work. It is important to note that this correlation is a function of Ca

_{L}and not Ca

_{B}like the others.

_{B}, with the deviation increasing for increasing Re

_{B}until a maximum of 10%. This behavior is due to the fact that the model equation is based on the correlation of Aussillous and Quere [11], valid mostly for low Ca

_{B}and Re

_{B}, to estimate the thickness of the liquid film [23].

_{B}.

_{B}region, not covered in the numerical simulations, the bubble velocity tend to the mean liquid velocity and, as already referred, the bubble tend to flow as a plug integrated inside the liquid.

_{B}number through the correlation of Aussillous and Quere [11], it is possible to roughly estimate the value of the Capillary number above which the bubble moves faster than the liquid in the center of the channel. Since for laminar flow the liquid in the center of the channel is flowing twice as fast as the average liquid velocity, the intended Capillary number, according to Figure 9, is around 0.74. This value should only serve as a rough estimate, mostly valid for low Re

_{B}(until Re

_{B}= 1).

_{I}), i.e., with a mean velocity equal to the bubble, and the total liquid flow rate (Q

_{T}) is plotted versus Ca

_{B}.

_{B}≤ 10, whatever the Ca

_{B}number, the curve is unique. All the fluid moves slower than the bubble for Ca

_{B}≥ 0.8. For Re

_{B}= 100 and Ca

_{B}= 0.8, the appearance of a toroidal closed recirculation region at the rear of the bubble (Figure 7) changes this behavior. This wake region travels at the bubble velocity and slows down the expansion of the liquid film at the rear of the bubble. This slow deceleration implies an acceleration of the liquid, in the flow direction, in front of the bubble.

#### 3.4. Simulations with Two Consecutive Bubbles

_{B}and Ca

_{B}studied, coalescence was only observed once, when Reynolds and Capillary numbers are simultaneously high (Ca

_{B}= 0.8; Re

_{B}= 100). The coalescence only happened for the shortest distance tested ( ${d}_{\mathrm{B}}$ = 61 μm). This pair of Re

_{B}and Ca

_{B}corresponds to the appearance of the closed recirculation wake at the rear of the bubble (Figure 7). This wake influences the reattachment of the velocity profile at the rear of the bubble and induces, near the center of the tube, regions with increased velocity. These regions are responsible by the accelerating of the following bubble.

_{B}= 0.1; Re

_{B}= 0.1) and another where no recirculation vortices are present (Ca

_{B}= 2; Re

_{B}= 0.1). The shape, thickness of the liquid film and velocity of the bubbles are identical to those for isolated bubbles, whatever the distance between the bubbles. As for the flow of an isolated bubble, the liquid for high Ca

_{B}is under “bypass flow” while for low Ca

_{B}the recirculation vortices between the bubbles are, as expected, no longer semi-infinite, forming a closed vortex.

## 4. Conclusions

_{B}and of 0.01–700 for Re

_{B}and the results are in good agreement with correlations and experimental results found in the literature. Outside of this range, a different approach is necessary, as spurious currents become more important for Ca

_{B}< 0.01 and 3D effects start to appear at higher Re

_{B}. However, for Ca

_{B}< 0.01, the results are easily extrapolated. For high Reynolds numbers and Ca

_{B}> 0.8, a recirculation region at the rear of the bubble appears, as in vertical slug macro studies.

_{B}and Re

_{B}. The velocity of the bubble, for low values of Ca

_{B}, is consistent with the stagnant film hypothesis, implicating that the bubble travels in the limit at the mean velocity of the liquid.

_{B}, the bubble moves at a velocity higher than the highest velocity on the liquid, and no recirculation regions appear.

_{B}until a plateau is reached, with its value being higher for higher Re

_{B}.

_{B}. If the thickness of the liquid film is unknown, other correlations are available, but these do not account with the effects of the Reynolds number. The correlations of Liu et al. [21] and Abiev and Lavretsov [22] were found to be more accurate for higher and lower Re

_{B}, respectively.

_{B}and Ca

_{B}(Ca

_{B}= 0.8; Re

_{B}= 100).

_{B}< 1000 and 0.02 < Ca

_{B}< 2.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

${A}_{B}$ | Cross section area of the bubble | m^{2} |

${A}_{\mathrm{film}}$ | Cross section area of the liquid film | m^{2} |

${A}_{\mathrm{MC}}$ | Cross section area of the microchannel | m^{2} |

D | Internal diameter of the tube | m |

${d}^{*}$ | Liquid film development length | m |

${d}_{B}$ | Distance between two consecutive bubbles | m |

$\overrightarrow{f}$ | Surface tension contribution | Pa·m^{−1} |

g | Acceleration due to gravity | m·s^{−2} |

k | Curvature of the interface | m^{−1} |

${L}_{\mathrm{s}}$ | Bubble length | m |

p | Pressure | Pa |

${Q}_{\mathrm{I}}$ | Volumetric flow rate of the liquid moving faster than the bubble | m^{3}·s^{−1} |

${Q}_{\mathrm{T}}$ | Volumetric flow rate of the liquid flowing in the tube | m^{3}·s^{−1} |

R | Internal radius of the tube | m |

$\overrightarrow{v}$ | Velocity vector | m·s^{−1} |

${V}_{B}$ | Taylor bubble velocity | m·s^{−1} |

${V}_{\mathrm{film}}$ | Mean velocity in the liquid film | m·s^{−1} |

${V}_{\mathrm{inlet}}$ | Mean velocity at the inlet | m·s^{−1} |

${V}_{\mathrm{L}}$ | Mean velocity in the liquid | m·s^{−1} |

${V}_{\mathrm{TP}}$ | Two-phase velocity | m·s^{−1} |

${V}_{\mathrm{wall}}$ | Velocity of the wall in the CFD model | m·s^{−1} |

x,r | Coordinates in the 2D simulation domain | m |

## Greek Letters

${\alpha}_{i}$ | Phase indicator (phase i) | |

${\alpha}_{\mathrm{C}}$ | Phase indicator (continuous phase) | |

${\alpha}_{\mathrm{D}}$ | Phase indicator (dispersed phase) | |

γ | Surface tension | N·m^{−1} |

δ | Annular liquid film thickness | m |

μ | Viscosity | Pa·s |

${\mu}_{\mathrm{C}}$ | Viscosity of the continuous phase | Pa·s |

${\mu}_{\mathrm{D}}$ | Viscosity of the dispersed phase | Pa·s |

${\mu}_{\mathrm{L}}$ | Viscosity of the liquid | Pa·s |

ρ | Density | kg·m^{3} |

${\rho}_{\mathrm{C}}$ | Density of the continuous phase | kg·m^{3} |

${\rho}_{\mathrm{D}}$ | Density of the dispersed phase | kg·m^{3} |

${\rho}_{\mathrm{L}}$ | Density of the liquid | kg·m^{3} |

${\rho}_{\mathrm{G}}$ | Density of the gas | kg·m^{3} |

## Dimensionless Groups

${\mathrm{Ca}}_{B}$ | Capillary number based on the velocity of the Taylor bubble |

${\mathrm{Ca}}_{\mathrm{L}}$ | Capillary number based on the mean velocity of the liquid |

$\mathrm{Eo}$ | Eötvös number |

$\mathrm{La}$ | Laplace number |

${\mathrm{Re}}_{B}$ | Reynolds number based on the velocity of the Taylor bubble |

${\mathrm{Re}}_{\mathrm{L}}$ | Reynolds number based on the mean velocity of the liquid |

${\mathrm{We}}_{B}$ | Weber number based on the velocity of the Taylor bubble |

## List of Acronyms

CFD | Computational fluid dynamics |

FFR | Fixed frame of reference |

MFR | Moving frame of reference |

PIV | Particle image velocimetry |

VOF | Volume-of-fluid |

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**Figure 8.**Contours of the bubble: (

**a**) Re

_{B}= 0.01; (

**b**) Re

_{B}= 0.1; (

**c**) Re

_{B}= 1; (

**d**) Re

_{B}= 10; (

**e**) Re

_{B}= 100; and (

**f**) Re

_{B}= 700. Thinner bubbles correspond to higher Ca.

**Figure 9.**Liquid film thickness as a function of Ca

_{B}. The solid lines represent the correlation of Han and Shikazono [17].

**Figure 12.**Ratio between the flow of liquid moving faster than the bubble and the total liquid flow against Ca

_{B}.

**Figure 13.**Contours of the bubbles, streamlines and velocity vectors for two consecutive bubbles. (

**a**) Ca

_{B}= 2; Re

_{B}= 0.1; (

**b**) Ca

_{B}= 0.1; Re

_{B}= 0.1

Mesh | Ca Range | Number of Elements | Size of the Elements |
---|---|---|---|

1 | ≥0.1 | 35,000 | 1 × 1 µm^{2} |

2 | >0.01 | 140,000 | 0.5 × 0.5 µm^{2} |

3 | =0.01 | 140,000 | 0.25 × 1 µm^{2} |

Ca_{B} | Re_{B} | V_{L} (m/s) | V_{B} (m/s) | V_{B}/V_{L} | δ/R |
---|---|---|---|---|---|

0.00983 | 0.982 | 0.0730 | 0.0822 | 1.13 | 0.0532 |

0.00990 | 9.90 | 0.231 | 0.262 | 1.14 | 0.0582 |

0.00983 | 98.2 | 0.730 | 0.822 | 1.13 | 0.0564 |

0.0100 | 699 | 1.93 | 2.21 | 1.14 | 0.0584 |

0.0303 | 0.0101 | 0.0119 | 0.0146 | 1.23 | 0.0988 |

0.0303 | 0.101 | 0.0376 | 0.0462 | 1.23 | 0.0984 |

0.0303 | 1.02 | 0.119 | 0.147 | 1.24 | 0.0982 |

0.0302 | 10.1 | 0.376 | 0.462 | 1.23 | 0.0988 |

0.0297 | 99.2 | 1.19 | 1.44 | 1.21 | 0.0918 |

0.0310 | 721 | 3.15 | 3.96 | 1.26 | 0.118 |

0.109 | 0.0109 | 0.0195 | 0.0290 | 1.48 | 0.180 |

0.110 | 0.110 | 0.0620 | 0.0919 | 1.48 | 0.180 |

0.101 | 1.10 | 0.196 | 0.290 | 1.48 | 0.179 |

0.108 | 10.8 | 0.620 | 0.904 | 1.46 | 0.174 |

0.103 | 103 | 1.96 | 2.73 | 1.39 | 0.154 |

0.110 | 773 | 5.19 | 7.73 | 1.49 | 0.222 |

0.301 | 0.0100 | 0.0260 | 0.0460 | 1.77 | 0.249 |

0.301 | 0.100 | 0.0823 | 0.145 | 1.77 | 0.248 |

0.350 | 1.17 | 0.296 | 0.534 | 1.81 | 0.256 |

0.293 | 9.76 | 0.824 | 1.41 | 1.72 | 0.238 |

0.270 | 89.9 | 2.61 | 4.12 | 1.58 | 0.207 |

0.823 | 0.0103 | 0.0372 | 0.0770 | 2.07 | 0.306 |

0.821 | 0.103 | 0.118 | 0.243 | 2.07 | 0.307 |

0.818 | 1.02 | 0.372 | 0.766 | 2.06 | 0.305 |

0.791 | 9.89 | 1.18 | 2.34 | 1.99 | 0.293 |

0.717 | 89.6 | 3.72 | 6.70 | 1.80 | 0.259 |

2.02 | 0.0101 | 0.0524 | 0.120 | 2.28 | 0.340 |

2.02 | 0.101 | 0.166 | 0.378 | 2.28 | 0.340 |

2.01 | 1.01 | 0.524 | 1.19 | 2.27 | 0.342 |

1.94 | 9.71 | 1.66 | 3.63 | 2.19 | 0.327 |

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**MDPI and ACS Style**

Rocha, L.A.M.; Miranda, J.M.; Campos, J.B.L.M.
Wide Range Simulation Study of Taylor Bubbles in Circular Milli and Microchannels. *Micromachines* **2017**, *8*, 154.
https://doi.org/10.3390/mi8050154

**AMA Style**

Rocha LAM, Miranda JM, Campos JBLM.
Wide Range Simulation Study of Taylor Bubbles in Circular Milli and Microchannels. *Micromachines*. 2017; 8(5):154.
https://doi.org/10.3390/mi8050154

**Chicago/Turabian Style**

Rocha, Luis A. M., João M. Miranda, and Joao B. L. M. Campos.
2017. "Wide Range Simulation Study of Taylor Bubbles in Circular Milli and Microchannels" *Micromachines* 8, no. 5: 154.
https://doi.org/10.3390/mi8050154