Siyuan Chen,Tao Zhang,Li Lv,Yanxiao Chen,Shengwei Tang

Low-Carbon Technology and Chemical Reaction Engineering Laboratory,School of Chemical Engineering,Sichuan University,Chengdu 610065,China

Keywords: Falling film Wavy microchannel CFD Mass transfer

ABSTRACT The flow in a liquid falling film is predominantly laminar,and the liquid-side mass transfer is limited by molecular diffusion.The effective way to enhance the mass transfer is to improve the liquid film flow behavior.The falling film behaviors of water,ethanol and ethylene glycol in nine different wavy microchannels were simulated by Computational Fluid Dynamics.The simulation results show that the falling film thickness exhibits a waveform distribution resulting in a resonance phenomenon along the wavy microchannel.The fluctuation of liquid film surface increases the gas–liquid interface area,and the internal eddy flow inside the liquid film also improves the turbulence of liquid film,the gas–liquid mass transfer in falling film microchannels is intensified.Compared with flat microchannel,the CO2 absorption efficiency in water in the wavy microchannel is improved over 41%.Prediction models of liquid film amplitude and average liquid film thickness were established respectively.

1.Introduction

As a new type of gas–liquid mass transfer device,the falling film microreactor has been attracted much attention due to its large specific surface area and ultra-thin liquid film with thickness less than 100 μm [1].Compared with conventional large-scale falling film equipment,the decrease of liquid film thickness can significantly increase the heat and mass transfer between gas phase and liquid phase.Furthermore,open falling film microchannels are capable of maintaining the flow pattern stability over a wide range of gas liquid flow rates,and it is conductive to the product separation of heterogeneous reaction [2].Therefore,falling film microreactors have been studied widely in gas–liquid organic chemical reactions such as fluorination [3–5],photocatalysis [6],chlorination [7]and selective hydrogenation [8,9].

Our research group has investigated the liquid film flow characteristics in falling film microchannels [10].Due to the limited microchannel size and fluid flow rate,the liquid flow inside the falling film microchannel is usually laminar.Thus,the mass transfer is dominated by molecular diffusion with poor efficiency.Many researchers suggested that the mass transfer resistance in the falling film microreactor is mainly from the liquid phase when the solubility of solute in liquid phase is small [11–13].Therefore,improving the liquid phase flow characteristics is crucial to enhance the mass transfer performance of falling film microdevices.

Improving the liquid flow behavior by changing the microchannel wall structure is one of the effective measures to enhance the gas–liquid mass transfer behavior.Ziegenbalget al.[2]set stagger herringbone grooves in the falling film microchannel to effectively increase the absorption rate of CO2in NaOH aqueous solution.It was found that the configuring staggered herringbone grooves at the bottom of the microchannels could disturb the bottom of the liquid film [14].Further work done by other researchers [15,16]indicated that the disturbance can reduce the concentration gradient of the liquid film and renew the gas–liquid interface with‘‘fresh”species.In our previous work,we also used stagger herringbone grooves to effectively intensify the isobutane/1-butene alkylation in the microreactor [17].By using micro-baffled plate,uniform fluctuations can also be made on the surface of the liquid film,and eddy flow are generated at the bottom of the liquid film to enhance the liquid-side mass transfer behavior [18,19].Chenet al.[20]enhanced the surface renewal of the liquid film by setting the micro-mixing zone in the double-sided falling film microchannel,and effectively improved the interphase mass transfer behavior.

It is an effective and feasible method to enhance the interphase mass transfer behavior through surface renewal and liquid film disturbance.In macroscopic large-scale falling film equipment,the wavy wall surface can disturb the liquid flow and enhance the gas–liquid phase mass transfer[21–23].Similarly,the wavy structure in the falling film microchannel should be able to improve the liquid flow characteristics,and thus intensity the interphase mass transfer.This paper aims to reduce the liquid side mass transfer resistance with enhancing liquid film disturbance and eddy by employing the wavy microchannel.

Due to the limitation of existing measurement techniques,it is difficult to accurately observe and quantitatively study the fluid flow and mass transfer characteristics in the falling film microchannel.The Computational Fluid Dynamics (CFD) method can be used to visualize the flow and mass transfer characteristics in the falling film microchannel.The CFD method can quantitatively describe the numerical solution of the fluid field and the concentration field in time and space,which has been successfully used to analyze the falling film flow and mass transfer behaviors in different falling film devices [10,24–26].

In this work,numerical simulation was used to investigate the liquid phase flow behavior and interphase mass transfer in different wavy microchannels.The effect of wavy structure parameters,fluid flow rates and liquid phase physical properties on the liquid film flow behaviors were investigated by using CFD simulation,which provide a theoretical basis for the design of new falling film microreactor.

2.Numerical Simulation

The VOF(Volume of Fluid)model was used to simulate the flow and mass transfer of various liquid phases in the wavy microchannels [27].This model is suitable for the simulation of gas–liquid two-phase free surface flow.The physical properties of the chemicals used in this study are listed in Table 1.

Table 1Physical properties of deionized water,ethanol and ethylene glycol (T=293.15 K,P=0.1 MPa)

Table 2Structural parameters of different wavy microchannel

2.1.Governing equations

The continuity equation for laminar flow is as follows:

The momentum equation is as follows:ulation results.When studying the two-dimensional flow of the liquid phase in the inclined wavy wall,Guet al.[31]found that surface tension is an important factor affecting the flow of the liquid film.We believe that the effect of surface tension on liquid film flow and mass transfer in the falling film flow can not be negligible.For a gas–liquid two-phase countercurrent operating system,the surface tension source term can be expressed using a CFS(Continuum Surface Force) model [32].

wherekdenotes the curvature of the gas–liquid interface,which can be defined by:

where ^n=n/|n|，n=?αL.In the mass transfer simulation,the VOF between 0.01 and 0.99 is considered as the gas–liquid mass transfer interface.

2.2.Boundary conditions and mesh generation

As shown in Fig.1,a two-dimensional wavy falling film microchannel model was established.A liquid-sealed structure is arranged at the bottom of the calculation domain in order to make the liquid phase flow out of the microchannel more smoothly,and the calculation is also easy to achieve stability [10,20].Furthermore,a 5 mm stabilizing section is provided above and below

wheret,p,u,g,Fvolare time,pressure,velocity vector,gravitational acceleration and additional volume force,respectively.In a gas–liquid two-phase system,ρ and μ in each calculation cell are the weighted average of the two-phase volume fraction,which are given by the following equations.

where the subscripts G,L represent the gas phase and the liquid phase in the calculation cell,respectively.The VOF model simulates the two-phase or multi-phase flow by solving the volume fraction of each fluid,thereby obtaining the phase interface between the immiscible fluids [28].The volume fraction equation is as follows:

Fig.1.Computational physics model and boundary conditions (scale in mm).

Due to the small size of the microchannel,surface tension has a large influence on fluid flow and gas–liquid interface.Tonget al.[29]and Shettyet al.[30]ignored the influence of surface tension on the flow of the liquid film,resulting in poor accuracy of the simthe wavy section in order to allow the liquid film to flow into and out of the wavy section more stably.According to Tonget al.[29],it was found that the film flow patterns and characteristics in the wavy section are very similar in each cycle starting from the second cycle when the channel length is longer than threefold wavelength.Therefore,five wavy cycles were set in the model for the liquid film flow in the periodic sinusoidal falling film microchannel.This not only can reduce the calculation time,but also accelerate the convergence speed.Only the simulation results from the three cycles in the middle were selected for further analysis.

As shown in the Fig.1,the wavelength,amplitude and waveform degree of each wavy period are λ,a,anda/λ,respectively.In order to investigate the effect of wavy size on liquid film flow,nine different sizes of wavy microchannels were provided in Table 2.

Fig.1 gave the physical model and boundary conditions.Both the gas phase inlet and the liquid phase inlet were set to the velocity entrance boundary condition.The gas phase outlet and liquid phase outlet were set to the pressure exit boundary conditions,and the outlet gauge pressure was set to zero.The wall was assumed to be a non-slip thermal insulation wall.The liquid inlet velocity was between 0.01 m·s-1and 0.02 m·s-1and the gas inlet velocity was zero.

The geometric model is built by using the Integrated Computer Engineering and Manufacturing code (ICEM),and the entire computational domain is divided by a quadrilateral mesh with high computational precision.Compared with the gas phase region,the momentum and concentration gradients in the liquid film flow region and the gas–liquid interface region are larger,so the mesh is denser in the liquid film flow region and the gas–liquid interface region,and the mesh in the gas phase bulk region is sparse.The velocity of liquid inlet and gas inlet are given,and the gauge pressure of outlet and gas outlet are set as zero.Wall and wavy microchannel are set as no slip constant temperature boundary condition.

Fig.3.The liquid film thickness vs grid number (Ethanol, VL=0.01 m·s-1, VG=0).

Generally,the meshing strategy needs to be optimized to balance the simulation accuracy and the computational effort.The simulation model was established with the structural parameters of the C wavy microchannel.Three different grid densities were used to discretize the computational domain for analyzing the grid independence.The local grid density of the three models is shown in Fig.2.The number of grids was 12,200,16,886,and 24,899,respectively.Anhydrous ethanol was used to investigate the effect of grid density on the simulation results.In order to determine the position of the liquid film,a gas–liquid interface is defined where the VOF is 0.5.Fig.3 shows the instantaneous thickness of the liquid film simulated with different mesh structures.It can be seen that the liquid film thickness in the wavy microchannels remains substantially unchanged despite the different grids number.However,the computation time of mesh III is much longer than the others.Therefore,mesh II was selected for further numerical simulation considering both accuracy and computation time.

2.3.Solution methods and convergence criteria

Fig.2.Diagrams of three different mesh densities.

Fig.4.Comparison of simulation results with experimental results.

Since the gas–liquid flow and mass transfer characteristics changed over time during the simulation,the transient simulation was used in this work.The high precision Geo-Reconstruct method was used to trace the gas–liquid interface.The Pressure Implicit Split Operator (PISO) was used for pressure–velocity coupling calculation,and PRESTO! Method was adopted for pressure discretization.Second-order upwind differencing was chosen for the solution of the momentum equations.The transient time step was between 10-6s and 10-5s.The simulation process was divided into two steps.First,a liquid film with a certain thickness was initially given,and then calculation started.After the velocity field reached the pseudo steady state,the position (VOF=0.5) was selected as the gas–liquid phase interface.The flow characteristics of the free liquid film surface in the middle of the wavy microchannels was studied accordingly.

2.4.Model validation

In order to validate the simulation accuracy,the experimental data in the literature [33]with a liquid flow of 2.413 m3·s-1·m-1was used to verify the simulation process.The calculation object was the semi-circular waveform surface structure (S-type surface structure in the literature) with a wave radius of 1.5875 mm and a wavelength of 6.35 mm.Silicone oil was used as the liquid medium.The model was solved to study the liquid film distribution on the middle-corrugated surface by ensuring the full development of the liquid film and avoiding the influence of entrance and exit,and the influence of the gas phase velocity on the liquid phase flow was neglected.

The liquid film distribution and film thickness simulation results were shown in Fig.4.It can be seen that the simulation results were in good agreement with the experimental data obtained in the literature,and the thickness deviation was less than 15%.Therefore,the established numerical model and simulation method was valid for investigating the liquid flow characteristics in the falling film channel.

3.Results and Discussion

3.1.Characteristics of wavy falling film

The fluctuating liquid film provides a higher gas–liquid interfacial area,and the eddy at the bottom of the liquid film also enhances the local mass transfer,thereby intensifying the liquid phase mass transfer behavior.Therefore,in order to optimize the wavy micro-structure and provide fundamental theory for the design,it is necessary to explore the relationship between the liquid film amplitude,average liquid film thickness,liquid phase flow rate,liquid phase physical properties and wavy structure parameters.

Fig.5.Liquid film thickness distribution in falling film wavy microchannel(Water,vL=0.01 m·s-1).

Fig.5 shows the liquid film position and thickness distribution of water in the F wavy microchannel.A regularly fluctuating gas–liquid interface was observed.The film amplitude (A) means the maximum change in the thickness of each liquid film,which is defined as the difference between the maximum value and the minimum value of the liquid film thickness in a cycle.Like the sinusoidal shape of the wavy wall,the free surface is also sinusoidal shape.This phenomenon is called resonance [34],which is caused by the interaction between the free surface and the wavy wall.Besides,Fig.5 shows the phase angle lag between free surface and wavy wall.This is because the liquid film in the wavy microchannel is affected by gravity,which results in inconsistency between the change of liquid film thickness and the wavy structure.Therefore,the phase difference(φ)of the free surface is introduced here to describe this phenomenon.TheDxmeans the distance in the X direction between the lowest point of the free interface and the lowest point of the wavy wall interface.φ is given by:

Liquid film thickness is one of the key hydrodynamic parameters for falling film flow.The reciprocal of film thickness can be regarded as the interface area in a unit liquid volume,which is an important parameter in the study of heat and mass transfer.The average film thickness was defined as the average of the liquid film thicknesses in the middle three cycles.

Fig.6.Effects of liquid phase properties on the free interface in the falling film wavy microchannel.

The effect of fluid physical properties on falling film free surface of F wavy microchannel is shown in Fig.6.Water,absolute ethanol,and ethylene glycol were used for simulation with an inlet velocity of 0.01 m·s-1,respectively.Obviously,liquid physical properties were closely related to film thickness.Water and ethanol have similar physical properties except surface tension,but the film thickness of water is thicker than that of ethanol in the same falling film microchannel.This indicates that the liquid surface tension has a significant effect on the thickness of the liquid film in the wavy falling film microchannel.This is different from the liquid film characteristics in the flat microchannel.Yanget al.[10]found that the film thickness formed by the anhydrous ethanol in the falling film microchannel is greater than the film thickness of the deionized water under the same experimental conditions.Furthermore,the film thickness of the ethylene glycol is thicker than that of the water due to the higher viscosity.It is found that the phase difference increases with increasing liquid surface tension,meaning that the water system has the largest one,and the ethanol system has the smallest one.

Figures of Figs.7–10 illustrate the effects of the wavy structure of microchannel on the film amplitude and the average film thickness.It is found that the structural parameters of the wavy microchannel have the similar effects on the average liquid film thickness and the liquid film amplitude.When the inlet flow rate and the liquid phase system are the same,both the liquid film amplitude and the average liquid film thickness increase with the increasing amplitude of the wavy microchannel,and decrease with the increasing of the wavelength of the wavy microchannel.Compared with water and ethylene glycol system,ethylene glycol with a higher viscosity accumulates more liquid in the trough of wave,so the liquid film amplitude and average liquid film thickness are larger.Moreover,the effects of wavy structural parameters have a greater influence on the liquid film amplitude and the average film thickness for the ethylene glycol system.However,the effects of wavy structure parameters on the film amplitude and the average film thickness of ethanol system are weaker than that of water and ethylene glycol system for their larger surface tension.Therefore,the liquid viscosity is the prominent factor affecting the liquid film amplitude and the average film thickness.In the same liquid system,higher liquid flow rate also leads to larger film amplitude and average film thickness.However,the effect of feed flow rate on the film amplitude and the average film thickness is much weaker than that of wavy structure parameters and fluid physical properties.Overall,the trend is that larger wave amplitude is associated with thicker liquid film.

Fig.7.Effect of amplitude of wavy microchannel on the amplitude of liquid film(λ=6 mm).

3.2.Liquid film amplitude prediction model

Based on the above results,a dimensionless model was derived to provide a quantitively predication tool for the liquid film amplitude of fluid in falling film microchannel.It is also useful to guide the design and optimization of the wavy falling film microchannel to strengthen the liquid phase mass transfer process.As discussed in the previous sections,many parameters,including structure parameters like wavelength (λ),amplitude (a) of the wavy microchannel,average film thickness (δ),the liquid flow rate (vL),liquid physical properties like viscosity (μ),density (ρ),surface tension (σ) and gravitational acceleration (g) affect the amplitude of wavy film (A).Consequently,seven dimensionless parameters includingA/δ,λ/δ,a/δ,Capillary numberCa,Weber numberWe,Froude numberFrand Bond numberBoare selected to study the hydrodynamic characteristics in FFMR.For simplification,WeandFrwere combined to giveBo.The Nomenclature shows the detailed formula.

According to the Rayleigh Dimensional Analysis theory,the dimensionless film thickness (A/δ) is given by

However,there are two dependent variables in Eq.(10),namelyAand δ which limit its utilization.In order to get a simplified model,the liquid film thickness (δflat) in the flat plate is used to replace δ,and the liquid film bulk flow rate (jL) is used to replace vL.δflatandjLare defined as following Nusselt theory.

Fig.8.Effect of wavelength of wavy microchannel on the amplitude of liquid film(a=0.4 mm).

Fig.9.Effect of amplitude of wavy microchannel on the average film thickness(λ=6 mm).

Fig.10.Effect of wavelength of wavy microchannel on the average film thickness(a=0.4 mm).

wheredis the inlet depth of the wavy microchannel,which is 0.2 mm.Eq.(10)is further simplified to get the following equation.

Fitting Eq.(13) with the simulation results by the least square method,the coefficients,c,f,h,i,j,o,q,are obtained and Eq.(13)is quantitively described as follows:

whereReranges from 1.4 to 8.1,Boranges from 1˙1×10-3to 0.017,Caranges from 0.029 to 0.11,a/λ ranges from 0.033 to 0.10.

Fig.11 reveals that CFD simulations results (ACFD) and the prediction results of Eq.(13)(AC)match well with a deviation of±20%,except for a few points where the liquid film amplitude is small.It is obvious that the established prediction model works well to predict the liquid film amplitude.

Eq.(14) quantitively illustrates the relationship between liquid film amplitude(A),structural parameters and properties of fluids.Ahas negative relationship with λ with an index of -3.31.Similarly,Ahas positive relationship withawith an index of 1.06 which is consistent with previous findings in this paper.In terms of the impact of fluid physical properties,Eqs.(11)and(12)indicate that δflatis proportional to ρ-2/3μ1/3,andjLis proportional to ρ-1/3μ-1/3.SoAis proportional to ρ-1.7μ0.15σ1.40,as shown in the following equation.

Since the fluid with larger surface tension and viscosity accumulates more liquid at the wall trough,a larger film amplitude is generated.However,the effect of inertial force in the microchannel is negligible,thereby the fluid density has little effect on the amplitude of the wavy film.

3.3.Average film thickness prediction model

The average liquid film thickness is one of the key parameters for falling film flow behavior.Similar to film amplitude,the parameters,including structure parameters like wavelength (λ) and amplitude(a)of the wavy microchannel,film thickness(δ),the liquid flow rate(vL),liquid physical properties like viscosity(μ),density (ρ),surface tension (σ),and gravitational acceleration (g),affect the average liquid film thickness of wavy film.Therefore,the functional relationship can be given by the following equation.

Fitting Eq.(16) with the simulation results by the least square method,the following equation was obtained.

whereReranges from 1.4 to 8.1,Boranges from 1˙1×10-3to 0.017,Caranges from 0.029 to 0.11,a/λ ranges from 0.033 to 0.10.

Fig.11.Wave amplitude of falling film: simulation results vs. model results.

Fig.12.Average film thickness of falling film: simulation results vs. model results.

Fig.12 reveals that the CFD simulations (δCFD) and the calculated film thickness by prediction model (δC) match well with a deviation of ±30%.Therefore,the established prediction model can be used to predict the liquid film thickness.Since the film thickness is affected by many parameters as discussed before,the trend of film thickness change is different in wavy microchannel and conventional microchannel.However,Ishikawaet al.[18]found that the average film thickness in the microchannel with micro-baffles was only affected by the microchannel structural parameters,while it was affected by both structural parameters of wavy microchannels and other flow parameters and fluid physical properties.The difference in microchannels with diverse micro-mixing structures should be the reason.

3.4.Internal flow characteristics

Mass transfer behavior is also intensified by the internal flow of the liquid film in the wavy falling film microchannel.Fig.13 shows the liquid phase steamlines in six different falling film microchannels with three liquids at a feed flow rate of 0.01 m·s-1.In the flat falling film microchannel,the surface is smooth and the liquid flows smoothly.In the A and I wavy microchannels with small waveform degrees,the liquid film does not form eddy flow at the wavy section.The possible explanation is that the low waveform degree results in a decreased pressure field with negligible internal fluctuation inside the liquid film,which makes it easy to form a continuous liquid film.On the other hand,the falling film generates eddy flow in C wavy microchannels with larger waveform degree.Furthermore,E wavy microchannel has the largest waveform degree,and also generates the largest eddy flow in both shape and number in the troughs despite the different test fluids.This is reasonable that the bulk liquid is hard to carry away the liquid in the troughs causing a back mixing over there with eddy flow inside.

The effect of fluid physical properties on the flow behavior in the E wavy microchannel are shown in Fig.13.Comparing with the streamlines of ethanol and ethylene glycol,the streamlines of water have much obvious eddy flow due to its polarity,low viscosity and high surface tension which are the dominate forces at micro-scale.This is consistent with the results of Tonget al[35].

3.5.Mass transfer behaviors

Fig.13.Liquid film flow pattern in different wavy microchannels (vL=0.01 m·s-1,red: liquid).

Using the geometric model and boundary condition settings in Fig.1,the effect of fluid flow behavior in the wavy liquid film on the gas–liquid mass transfer was explored.The mass transfer simulation method was the same as the literature[20].The gas phase inlet width was set as 2 mm.The gas and liquid feed flow rates were 0.03 m·s-1and 0.01 m·s-1,respectively.The mass transfer performances of CO2absorption in water were investigated in flat falling film microchannel and A,B and C wavy falling film microchannels.The vertical gas–liquid contact distance was 40 mm.When the gas–liquid interphase mass transfer calculation was stable,the mass fraction of CO2at the liquid phase outlet was recorded,as shown in Fig.14.

Fig.14 shows that the liquid outlet concentration in the wavy microchannel is increased by about 41%compared with that in flat microchannel.This is that we expected.Because the wavy structure can generate eddy flow inside the liquid film,the local mass transfer rate can be improved.Furthermore,the fluctuation of the liquid film increases the gas–liquid phase mass transfer interface area and also increases the gas–liquid contact time.

Fig.14.Liquid concentrations in different microchannels.

Fig.15.Velocity distribution of liquid film in different microchannels.

Fig.16.Gas-liquid mass transfer rate distributions in different microchannels.

The mass transfer intensification mechanism was further investigated at the gas–liquid interface (VOF=0.5) in different microchannels.Figs.15 and 16 indicate the flow velocity distribution and the gas–liquid mass transfer rate distribution.When the liquid film flows through the wavy section,the surface of the liquid film and flow velocity fluctuates,thereby causing a difference of mass transfer rate between flat and wavy microchannels.However,the liquid film flows smoothly in the flat microchannel,so the gas–liquid mass transfer rate does not change much.As the falling film distance increases,the mass transfer rate decreases.This is reasonable that the liquid concentration increases along the flowing path,resulting in reduced mass transfer driving force,and finally reduce the inter-phase mass transfer rate.

4.Conclusions

The falling film flow in the designed wavy microchannels were investigated by CFD simulation.The effects of wave structure parameters,feed flow rate,viscosity and surface tension on the liquid flow behaviors,film amplitude and average film thickness were investigated in detail.

The liquid film surface resonates with the wavy wall surface in the wavy falling film microchannels,thereby forming a regular wavy liquid film.Fluid physical properties have different contribution to the liquid film flow behaviors.The eddy formation and phase difference inside the wavy film are mainly affected by surface tension,and the film amplitude and average film thickness are mainly affected by viscosity.

Two simple prediction models have been established for film amplitude and average film thickness respectively according to the simulation results.These models integrate the influence of gravity,surface tension,viscous force and wavy structure parameters.The model prediction is reasonably accurate with a standard deviation of ±20% and ±30% for film amplitude and average film thickness respectively.

Compared with flat microchannel,the simulation results show that the CO2absorption efficiency in water is improved over 41%for the wavy microchannel,which reveals that the wavy microstructure can enhance the gas–liquid mass transfer significantly.

Nomenclature

Awave amplitude of falling film,m

awave amplitude of wavy microchannel,m

BoBond number,Bo=ρ·g·δ2/σ=We/Fr

bwidth of microchannel,m

CaCapillary number,Ca=μ·vL/σ

ddepth of microchannel,m

e number of microchannel

Fσsurface tension source term,N·m-3

FrFroude number,Fr=/（g·δ）

g gravitational acceleration,m·s-2

jLaverage velocity in falling film,m·s-1

llength of the falling film in simulation,m

nsurface normal

Qflow rate,m3·s-1

Rfitting correlation coefficient

vLliquid feed rate,m·s-1

WeWeber number,We=ρ··δ/σ

α volume fraction

δLthe thickness of falling film,m

λ wavelength of wavy microchannel,m

φ free surface phase difference

κ curvature of free surface

δ the average thickness of falling film,m

δflatthe liquid thickness on flat falling film microchannel,m

μ viscosity,Pa·s

ρ density,kg·m-3

σ surface tension coefficient,N·m-1

Subscripts

G gas phase

L liquid phase

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No.21576168).

Supplementary Material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cjche.2020.09.014.