Lin Sheng,Yuchao Chen,Jian Deng,Guangsheng Luo*

The State Key Laboratory of Chemical Engineering,Department of Chemical Engineering,Tsinghua University,Beijing 100084,China

Keywords:Bubble swarm Flow ideality Viscous fluids Gas-liquid microreactor

ABSTRACT The flow ideality of bubbly microflow remains unclear even though it is vital for the design of microreactors,especially the ideality of bubble swarm microflow for large-scale gas-liquid microreaction processes.This work is the first time to report the ideality analysis of the microbubble swarm in a relatively large microchannel.The bubble swarm microflow has undergone two conditions: quasihomogeneous plug flow and liquid phase/gas-liquid quasi-homogeneous phase two-phase laminar flow.Both the deviations of void fraction and bubble velocity from the ideal plug flow can divide into two parts,and the two transition points simultaneously happen at the velocity ratio of 1.25.There exists a critical capillary number to maintain the quasi-homogeneous plug flow,which could be regarded as the general laws for the design of gas-liquid microreactors.Finally,a novel model is developed to predict the bubble velocity.This work could be very helpful for the large-scale gas-liquid microreactors design.

1.Introduction

Microreactor,with the excellent performances in heat and mass transfer,precise time control with uniform reaction conditions,and inherent safety aspects,has attracted considerable attention in various fields,such as chemical reaction [1-3],material synthesis [4-6],solvent extraction [7,8],and gas absorption [9-11],etc.Since the effect of the surface (interfacial)tension plays a significant role in multiphase microflows,it has been confirmed that the hydrodynamics of the two-phase flow in microfluidic devices differ from those in conventional channels with centimeter dimensions [12-16].Therefore,the design rules of the conventional reactor are no longer suitable for the microreactor design to a certain degree.In order to design and fabricate a microreactor effectively for the practical industrial application,the characteristics of the gas-liquid microflow including the bubble size,the void fraction of the gas phase,and the relative velocity of the two phases in the microchannel should be well understood.Besides,the reaction kinetics is the most fundamental data for reactor design,and many previous works have reported that more intrinsic reaction kinetics have been obtainedviathe microreaction technology for the reason that the mass transfer resistance is eliminated to some extent when compared with the traditional batch reactor,and the reaction time used in many previous reports about heterogeneous reaction just simply chose the average residence time of the multiphase system [17-21].However,it is noted that the actual residence time for the gas and liquid phase in the microreactor may be different due to the non-ideality of the system,and the hypothesis of ideal plug flow in many microflow conditions is not correct.The true reaction time should be modified to obtain the intrinsic reaction kinetics.Therefore,in terms of the microreactor design and intrinsic reaction kinetic acquisition,the flow ideality and general laws of the gas-liquid two-phase flow in the microreactor should be urgently analyzed.

To the ideality analysis of the gas-liquid system in terms of the void fraction,Armand [22] reported that there exists a linear relationship between the void fraction and the volumetric quality of the gas phase,and this conclusion has been further verified in the air-water flow for six different orientations when the gas absorption is negligible,where the coefficient is 0.8[23].However,when the gas absorption is remarkable in the gas-liquid system,the linear relationship is no longer suitable.Yinet al.[16]observed the hydrodynamics of the CO2bubble flows in the MEA aqueous solutions mixed with ionic liquids in a microchannel,and they concluded that the coefficient of 0.8 should be modified with the Hatta number.Except for the linear relationship,the nonlinear relationship has also been reported previously.Chunget al.[12] investigated the N2-water microflow with the volumetric quality ranges from 0 to 1.0,and they proposed a nonlinear prediction modelviathe mathematical analysis of the experimental data.Further,Xionget al.[24] found that the coefficient in the nonlinear model developed by Chunget al.[12] is not a constant value in different microchannels and the coefficient is determined by the hydraulic diameter of the microchannel.

To the ideality analysis of the gas-liquid system in terms of the bubble velocity,Fukanoet al.[25]chose air and water as the experimental system,and they reported that the bubble velocity is proportional to the superficial velocity of the gas-liquid two phases.Further,Mishimaet al.[26] investigated the air-water flow in a capillary tube with different inner diameters and pointed out that the coefficient varies with diameter in an exponential relationship,and they developed a modified model to predict the bubble velocity based on the model proposed by Fukanoet al.[25].However,Choiet al.[27]found that the bubble velocity in the N2-water slug flow regime is almost equal to the two-phase superficial velocity when the aspect ratio of the microchannel is 0.47.In addition to the linear relationship,the nonlinear relationship between the bubble velocity and superficial velocity has also been reported.Liuet al.[28] concluded that the ratio of the air bubble velocity to the superficial velocity can be well quantified by the capillary number of the two phases.The prediction models including the Bond number,Reynolds number,and Weber number have also been developed [29,30],and their working system is composed of N2,water,and ethanol.

Above all,the majority of the prediction models for the deviation of the void fraction and bubble velocity from the ideal-plug flow were carried out in the low viscous fluids.However,highly viscous solutions are widely encountered in food and polymer engineering,which attracts increasing attention [31-34],but the fundamental research on the flow ideality of the highly viscous gas-liquid system is rarely reported.Importantly,to realize the practical industrial application of the microchannel reactor,the numbering-up method has been successfully proposed to promote the throughput of the microdevice.However,just simply choosing the numbering-up method for the gas-liquid system with higher viscosity is not suitable in terms of the higher flow pressure drop,thus a suitable size enlargement method is needed for this special system to increase its throughput.It is worth noting that the bubble/droplet swarm flow is the actual flow state when the bubbles/-droplets flow in a relatively large channel rather than the flow state that a string of bubbles/droplets flow when the bubbles/-droplets flow in a relatively small channel,and the phenomenon of bubble/droplet swarm flow is the same as the bubbles/droplets flow state in the collection channel under the numbering-up method [8,35-37].Therefore,the existing models for a string of bubbles flow in a relatively small microchannel may not be suitable for the bubble swarm microflow in relatively large channels.In general,considering the research blank of the viscosity effect and the bubble swarm microflow in a relatively large microchannel,different highly viscous glycerol-water solutions were used to explore the ideality of the bubble swarm microflow in terms of the void fraction and bubble velocity.The effects of the operation conditions on the void fraction and volumetric quality were first studied,and then the influences of operation conditions on the bubble velocity were investigated.Besides,the relationship between the bubble velocity and void fraction was revealed.Finally,a novel mathematical model was proposed to predict the bubble velocity.This work has tried to fill the research blank about the ideality analysis of bubble swarm microflow in the viscous gas-liquid system and provide new fundamental understandings and general rules for the design of large-scale gas-liquid microreactors.

2.Materials and Methods

2.1.Experimental facilities

The experimental setup for the flow ideality analysis is demonstrated in Fig.1.The liquid was pumped into the microdevice by a high-pressure syringe pump(LSP01-1BH,LongerPump,China),and the gas phase was delivered from an N2cylinder and controlled by a constant-pressure controller (OB1,Elveflow,France),where the injection pressure is gauge pressure.The microbubble was formed by a capillary embedded step T-junction,and the geometrical parameters and fabrication method are the same as the microdevice thatweused inourpreviouswork[38].Thebubbleswarmmicroflow was easily formed by an expansion of the microchannel size after the bubble generation[39,40].The gas-liquid microflow was recorded by a high-speed camera(i-SPEED TR,Olympus,Japan),and the frequency for the capture of the images was fixed at 2000 fps(frames per second) in all the experiments.Once the operation conditions of the two phases were set,5 min were allowed for the system to stabilize.All the experiments were carried out under 25°C and atmospheric pressure,and the experimental temperature is ensuredviaa constant temperature ventilation system.

2.2.Reagents and materials

Four glycerol-water solutions with different mass fractions were used as the liquidphase.Sodium dodecylsulfatewith a mass fraction of 0.3%was added into the liquid phase to decrease the interfacial tension and prevent the coalescence of the bubbles.N2was chosen as the gas phase.The physical properties of various liquid phases are listed in Table 1.The density of the liquid (ρ) was determinedviaa densimeter(LC-MDJ-600G,LICHENTechnology,China).The liquid viscosity(μ)was measured by a viscometer(DV-IIa+P,Brookfield,USA).The interfacial tension (σ) was obtained by a tensiometer (OCAH200,DataPhysics Instruments,Germany).The capillary number of the liquid phaseCaL(CaL=μLvL/σ)ranged from 0.01 to 0.11.The Weber number of the liquid phaseWeL(WeL=ρLv2LDh/σ)and the gas phaseWeG(WeG=ρGv2GDh/σ)varied from 0.01 to 0.96 and 0.38×10-7to 0.14×10-3,respectively,vLand vGrepresent the superficialvelocity of the liquidphase andgas phase in the test microchannel(W×H:1100 μm×365 μm),respectively.

Table 1 Physical properties of various liquid phases (25 °C)

Table 2 Correlations on the void fraction in the literature

Fig.1.Schematic diagram of the experimental setup.

Fig.2.Flow behavior of the bubble swarm under different operation conditions(PG=100 kPa,μ=17.5 mPa·s,1-12:QL=500,600,700,800,1000,1200,1400,1600,1800,2000,2300,2600 μl·min-1).

2.3.The calculation method

In this experiment,the gas phase volume (VG)was equal to the product of the bubble numbers and volume in the test channel,and the total volume of the test channel (VT) was fixed (L×W×H:2134 μm × 1100 μm × 365 μm),whereLrepresents the length of the observation window of the microscope,as shown in Fig.1.Therefore,the void fraction of the gas phase (α) was equal to the ratio ofVGtoVT[16].The volumetric quality of the gas phase (β)was equal to the ratio of the gas flow rate(QG)to the total flow rate of the two phases(QG+QL),and the gas flow rate was equal to the product of the bubble volume and formation frequency.The bubble velocity was calculated by the movement distance and time.

3.Results and Discussion

3.1.Flow behavior of bubble swarm in the microchannel

To the flow behaviors of the bubble swarm in the microchannel,Garsteckiet al.[39] and Ravenet al.[40] have reported that the flow pattern of the bubble swarm could be developed into a highly regular flowing crystal that only a very thin liquid film exists between the adjacent bubbles when the bubble size or the void fraction is relatively high,and the flow behavior of this flow pattern has been reported in their works.Therefore,we don’t further investigate this flow pattern in this work.In this experiment,the void fraction and the bubble size are smaller than 0.6 and 250 μm,respectively.It means that the bubble size is much smaller than the microchannel size.Thus,the bubble swarm microflow in this work has been easily realized.The typical flow patterns of the N2bubble swarm microflow in the viscous glycerol-water solution at a fixed gas injection pressure and liquid viscosity are demonstrated in Fig.2.From Fig.2,we can find that the bubbles are crowded and closely contact each other at a lower liquid flow rate,which leads to the morphology of some bubbles being irregular,and the bubbles occupy a majority of the microchannel along the width direction.Especially,when the liquid flow rate is smaller than 600 μl·min-1,the periphery of the bubble swarm is very close to the microchannel wall.Due to the relatively larger bubble size at a lower liquid flow rate,the crowded bubbles also occupy a majority of the microchannel along the height direction.However,as the liquid flow rate further increases,the bubbles are gradually scattered and prefer to flow in the center of the microchannel,which is mainly for the reason that the bubbles will be rearranged and self-assembled in a confined zone when the void fraction is relatively higher [41].And this phenomenon will be gradually disappeared with the decrement of the void fraction,and the bubbles’behavior is gradually dominated by the velocity profile of the liquid phase.

3.2.Flow ideality analysis in terms of the void fraction

In this section,we investigated the influences of operation conditions on the void fraction (α) and volumetric quality of the gas phase (β),as depicted in Fig.3,where the solid dot represents the void fraction and the hollow dot symbols the volumetric quality.From Fig.3(a),we can see that both the void fraction and volumetric quality are gradually decreased with the increment of the liquid flow rate at a fixed gas injection pressure and liquid viscosity,and the decrement rate is gradually decreased,which indicates that the void fraction is more insensitive to the variation of the liquid flow rate at a smaller void fraction.It could be for the reason that the bubbles tend to flow in the microchannel center with the decrement of the void fraction(see Fig.2) and only a minority of the liquid phase interacts with the bubbles,which leads to insensitivity.At a given liquid flow rate and liquid viscosity,both the void fraction and volumetric quality are obviously increased as the gas injection pressure increases,as shown in Fig.3(a),which primarily results from that the gas flow rate increases with the gas pressure [38].Besides,the liquid viscosity effects were also explored as demonstrated in Fig.3(b).Both the void fraction and volumetric quality are dramatically decreased with the increase of the liquid viscosity when the liquid flow rate and gas injection pressure are fixed,and this result agrees with the conclusion reported by Kawaharaet al.[29]for the gas-liquid slug microflow.It is worth noting that the decrement rates of the void fraction and volumetric quality are increased as the viscosity of the liquid phase increases,which implies the importance of the viscosity effect on the variation of the void fraction.

Fig.3.(a) The effects of the gas injection pressure and liquid flow rate on the void fraction and volumetric quality,and(b) the effects of the liquid viscosity and liquid flow rate on the void fraction and volumetric quality.

Fig.4.(a)Variation of the deviation between the volumetric quality and void fraction with different operation conditions,(b)the distribution of CaL under various Q*and the relationship between the gas pressure and capillary number.

Interestingly,it can be observed that the void fraction is always smaller than the volumetric quality when the operation conditions are the same in all the experiments,which means the residence time of the two phases cannot be described as the ideal plug flow.Herein,we tried to explore a deep relationship between the void fraction and volumetric quality through the investigation of the difference between the two parameters,as shown in Fig.4(a).We can observe that the deviation between the void fraction and volumetric quality with the liquid flow rate can be divided into two stages: (Ⅰ) it is first increased to the maximal value,and then(Ⅱ) it is gradually decreased as the liquid flow rate increased at a given liquid viscosity and gas injection pressure.Here we define the transition liquid flow rate between the two stages asQ*,as the vertical dash line presented in Fig.4(a).TheQ* means that the value of β-α has reached its maximum at this liquid flow rate under a fixed gas injection pressure and liquid viscosity.The value ofQ* is increased with the increment of the gas injection pressure at a fixed liquid viscosity and decreased as the liquid viscosity increases when the gas injection pressure is the same.That is to say,the liquid viscosity and gas injection pressure has an opposite effect on the deviation of the void fraction.From the insert picture in Fig.4(a),we can see that the variation rates of the void fraction deviation in both stages are increased with the increment of liquid viscosity,thus the importance of the viscosity effect on the variation of the void fraction is further confirmed.To better understand the viscosity effect,we further calculate the capillary number of the liquid phase at variousQ*and find that the values of transition capillary number under various liquid viscosity almost are the same when the gas injection pressure is fixed as the insert picture shown in Fig.4(b),which implies that there exists a critical capillary number to guide the microreactor design and operation for large-scale gas-liquid microreactors.Importantly,we also find that the critical capillary number under different gas injection pressure strictly follows a linear relationship with the gas injection pressure(see the fitted line in Fig.4(b)).When the capillary number of the liquid phase is larger than the critical capillary number,although the absolute deviation of the void fraction (β-α) is gradually becoming small as shown in Fig.4(a),the relative deviation of the void fraction(α/β)is significant(see Fig.11),which implies that the ideality of the gas-liquid system is becoming worse and unacceptable,and this phenomenon should be strictly avoided in the design and application of microreactor.

Indeed,some previous works have concluded that the void fraction could be predicted by the linear and nonlinear models based on the volumetric quality of the gas phase,and the prediction models with their corresponding flow regimes are listed in Table 2.To verify whether the existing models for a string of bubbles in relatively small microchannels are suitable for the bubble swarm microflow in viscous fluids in this work,we chose the volumetric quality as the horizontal coordinate and its corresponding void fraction as the vertical coordinate to plot the comparison map as shown in Fig.5(a).We can see that the prediction models in Table 2 are failed to correlate the void fraction and volumetric quality for the bubble swarm microflow in viscous fluids in a wide range of volumetric quality.Further,we have systematically compared the deviations of the predicted void fraction in different prediction models,as demonstrated in Fig.5(b).We can find that the predicted deviation is obvious in all the four models when the volumetric quality of the gas phase is smaller than 0.25,and the two linear models [22,23] can well predict the void fraction to some extent when the void fraction is larger than 0.25.However,it is noted that the predicted deviation of the linear models has a nonlinear relationship with the volumetric quality.Thus,a novel nonlinear prediction model for the bubble swarm microflow in viscous fluids should be developed to precisely predict the void fraction based on the volumetric quality for a wide range (it will be discussed in Section 3.4).

Fig.5.(a) Comparison between the experimental data and prediction models in Table 2,and (b) the deviation of the predicted void fraction of the models in Table 2.

3.3.Flow ideality analysis in terms of the bubble flow velocity

In the above section,we have mentioned that the bubble flow behavior may have a significant influence on the deviation between the void fraction and volumetric quality.Herein,we investigated the effects of operation conditions on the bubble velocity in this section,as shown in Fig.6.The solid dot represents the average bubble flow velocity (vb),which is a statistic value based on lots of bubbles (at least 30 bubbles at different locations of the test channel).The hollow dot symbols the superficial velocity of the two phases(vsup),which equals the ratio of the total flow rate of the gas and liquid phases to the cross-section area of the test microchannel (W×H: 1100 μm × 365 μm).As depicted in Fig.6(a),at a given gas injection pressure of 150 kPa and liquid viscosity of 45.6 mPa·s,the bubble velocity is first slightly increased when the liquid flow rate ranges from 400 μl·min-1to 700 μl·min-1,but the bubble velocity is abruptly increased with a large increment rate when the liquid flow rate exceeds 700 μl·min-1,and this phenomenon is generally exited in all the operation conditions in terms of the gas pressure variation and the liquid viscosity variation (see Fig.6).Besides,we can see that the bubble velocity is increased with the increment of the gas injection pressure and decreased as the liquid viscosity increases when the liquid flow rate is relatively small,but there is almost no obvious difference in bubble velocity under various gas injection pressure and liquid viscosity when the liquid flow is relatively high.However,Kawaharaet al.[29] concluded that the bubble velocity is increased with the increase of the liquid viscosity for the reason that liquid film thickness around the bubble decreases with the liquid viscosity in the slug flow regime.The two opposite conclusions mainly result from that the pure liquid region around the bubble swarm in this microchannel is increased with the increment of the liquid viscosity under the constant pressure injection method in this viscous system,as shown in Fig.7.Generally,the flow behavior of the bubble swarm has undergone two different conditions with the increment of the liquid flow rate: quasihomogeneous plug flow and pure liquid phase/gas-liquid quasihomogeneous phase two-phase laminar flow.From Fig.6,we also can observe that the superficial velocity is almost linearly increased as the liquid flow rate increases,it is mainly for the reason that the gas flow rate is decreased with the increment of the liquid flow rate under the constant pressure injection method of the gas phase [33].

Fig.6.(a) The effects of the gas injection pressure and liquid flow rate on vb and vsup,and (b) the effects of the liquid viscosity and liquid flow rate on vb and vsup.

Fig.7.The influence of the liquid viscosity on the bubble flow behavior(PG=100 kPa).

Interestingly,the bubble velocity is always larger than the superficial velocity of the two phases when the operation conditions are the same in all the experiments.Similar to the investigation of the deviation of the void fraction in Section 3.2,we also tested the deviation between the bubble velocity and superficial velocity and tried to find a deep relationship between the two parameters in this part.Accordingly,we have plotted the variation of the velocity deviation with different operation conditions,as depicted in Fig.8(a).We can see that the bubble velocity deviation is increased with the increment of the liquid flow rate but decreased with the increment of the gas injection pressure.The reason could be explained as follow.Research has reported that the Taylor bubble flow velocity deviation is mainly for the liquid leakage flow between the Taylor bubble and rectangular channel wall(liquid flow region)[26].A smaller liquid flow region between the bubble and channel wall is beneficial to the ideal plug flow state,and the ratio of the bubble velocity to the superficial velocity is gradually close to 1.00 [27].In this study of bubble swarm flow,the pure liquid flow region between the bubble swarm and channel wall is increased with the increment of its flow rate but decreased as the gas injection pressure increases,thus the velocity deviation is more obvious under a lower gas injection pressure and higher liquid flow rate.Besides,the bubble velocity deviation is increased as the liquid viscosity increases,as shown in the insert picture in Fig.8(a),and it is for the reason that the pure liquid flow region is increased with the increment of the liquid viscosity when the gas injection pressure and liquid flow rate are the same (see Fig.7).Importantly,we can see that the increment rate of the bubble velocity deviation also exits a turning point as the liquid flow rate increases.In the first condition,the increment rate of the bubble velocity deviation keeps at a relatively lower degree,as the gray dash line depicted in Fig.8(a).And then the growth rate is abruptly increased with the increment of liquid flow rate in the second condition,thus we defined the transition liquid flow rate asQ**,as the vertical dash line presented in Fig.8(a).TheQ**means that the value of vb-vsuphas finished its low increment stage at this liquid flow rate under a fixed gas injection pressure and liquid viscosity.To illustrate the two conditions more vividly,we chose the transition liquid flow rate of 1300 μl·min-1as an example when the gas injection pressure and liquid viscosity are 100 kPa and 17.5 mPa·s,respectively,as depicted in the insert picture in Fig.8(a).When the liquid flow rate ranges from 500 μl·min-1to 1300 μl·min-1,the width of the pure liquid region is smaller than a bubble diameter(see Fig.2),which indicates that the interaction between the bubble swarm and the liquid phase is strong and they will flow together at about the same speed to some extent,and this flow pattern could be regarded as the quasi-homogeneous plug flow.However,when the liquid flow rate exceeds 1300 μl·min-1,the pure liquid region is very obvious(see Fig.2),which implies that a proportion of the liquid phase near the channel wall has no influence on the flow behavior of the bubble swarm.Thus,we defined this flow pattern as the liquid phase/gas-liquid quasi-homogeneous phase two-phase laminar flow.That is to say,when the void fraction is relatively large,the gas-liquid quasi-homogeneous microflow happens in the whole channel,but the gas-liquid quasihomogeneous microflow only happens in the microchannel center when the void fraction is relatively small.Therefore,the definition of the two conditions of the bubble velocity deviation is reliable,and the transition liquid flow rate (Q**) is increased with the increase of the gas injection pressure but decreased as the liquid viscosity increases,as shown in Fig.8(a).

Surprisingly,we can find that all the ratios of the bubble velocity to the superficial velocity at differentQ**are close to 1.25 within±5%,as demonstrated in Fig.8(b).That is to say,when the ratio of the bubble velocity to the superficial velocity is smaller than 1.25,the bubble velocity deviation is increased at a relatively slow rate,thus we regard this flow range as the quasi-homogeneous plug flow.Accordingly,when the velocity ratio is larger than 1.25,the bubble velocity deviation is increased at a relatively large rate,which indicates that the gas-liquid microflow significantly deviates from the ideal plug flow,and this phenomenon is what we should strictly avoid in the design and application of microreactor.Although a large microchannel can simultaneously enhance the throughput and reduce the pressure drop when compared with the microchannel with a smaller size at a suitable void fraction range,the expansion microchannel has no effect on flow behavior change when the void fraction is very small.Generally,the microchannel size design is a relatively complex task,and the size of the microchannel is neither as small as possible nor as large as possible.

Fig.8.(a) The effects of the operation conditions on the deviation of bubble velocity,and (b) the distribution of vb/vsup under various Q**.

As shown in Figs.4(a)and 8(a),the variations of the deviation of void fraction and bubble velocity with the liquid flow rate all exist a transition point at a given liquid viscosity and gas injection pressure,thus we compared their corresponding liquid flow rate as shown in Fig.9,where the vertical coordinate and horizontal coordinate of each dot symbolize the transition liquid flow rate of the bubble velocity deviation and void fraction deviation,respectively.Delightedly,we find that the transition points of the void fraction deviation and bubble velocity deviation almost happen at the same liquid flow rate(Q*=Q**)when the gas injection pressure and liquid viscosity are the same,which indicates that the variations of the void fraction deviation and bubble velocity deviation have a very strong correlation.Generally speaking,the value ofQ* andQ**could be regarded as the critical operation parameter to keep the flow ideality of the gas-liquid system,which means that the maximal liquid flow rate should be smaller than theQ* under a given gas injection pressure.Otherwise,the flow ideality of the gas-liquid system will significantly deviate from the ideal plug flow.

Actually,some previous investigations have reported that the bubble velocity could be predicted by the linear and nonlinear models based on the superficial velocity of the gas-liquid two phases,and the mathematical models with their corresponding flow regimes are shown in Table 3.From Table 3,we can see that the coefficient of the linear model just depends on the geometrical parameter,but the coefficient of the nonlinear model is determined by many factors,which includesCanumber (the ratio of the viscous shearing force to the interfacial tension force),Wenumber (the ratio of the inertial force to the interfacial tension force),andBonumber (Bo=（ρL-ρG）g/σ,the ratio of the gravity force to the interfacial tension force).We chose those models to predict bubble velocity in this work,as shown in Fig.10.We can find that all the linear models are failed to predict the bubble velocity,and the nonlinear model of Liuet al.[28]can well predict the bubble velocity when the liquid flow rate is smaller and the nonlinear model of Kawaharaet al.[29] can partially predict the bubble velocity when the liquid flow rate is higher.Herein,a novel nonlinear mathematical equation for the bubble velocity of the bubble swarm microflow is also urgently needed to develop(it will be discussed in Section 3.4).

Fig.9.The transition liquid flow rate comparison between the bubble velocity deviation and void fraction deviation.

Table 3 Correlations on the bubble velocity in the literature

3.4.Prediction models for the bubble velocity and void fraction

As mentioned in Sections 3.2 and 3.3,the existed dimensionless prediction models for the bubble velocity and void fraction are not suitable for the bubble swarm microflow in the viscous gas-liquid systems.Therefore,we tried to deeply explore another relationship between the dimensionless bubble velocity and void fraction in this section,as demonstrated in Fig.11,where the solid dot represents the ratio of the bubble velocity to the superficial velocity(vb/vsup) and the hollow dot symbols the ratio of the void fraction to the volumetric quality (α/β).From Fig.11(a),we can see that the ratio of the bubble velocity to the superficial velocity is increased with the increment of the liquid flow rate.It is noted that the increment rate is smaller in a larger gas injection pressure condition,which leads to the ratio of the bubble velocity to the superficial velocity is decreased as the gas injection pressure increases,as shown in Fig.11(a).Besides,the ratio of the bubble velocity to the superficial velocity is dramatically increased with the increment of the liquid viscosity when the liquid flow rate and gas injection pressure are fixed as depicted in Fig.11(b),and this phenomenon is consistent with the results reported by Kawaharaet al.[29].Importantly,the ratio of the bubble velocity to the superficial velocity is always larger than 1.0 in all experiments.From Fig.11,we also find that the variation of the ratio of the void fraction to the volumetric quality with different operations is totally opposite to the variation of the ratio of the bubble velocity to the superficial velocity,and the ratio of the void fraction to the volumetric quality is always smaller than 1.0 in all experiments.Further,we have investigated the variation of the products of the values of vb/vsupand α/β with different operation conditions,as the densely dot demonstrated in Fig.12,all the products of vb/vsupand α/β are close to 1.0 within ±5%,which implies that the experiment design and calculation method of this work is reliable due to the mass conservation.

Due to the strong relationship between the dimensionless bubble velocity and void fraction,the void fraction can be easily obtained via the ratio of the bubble velocity to superficial velocity,thus only one prediction model based on the operation parameters is needed to be developed to obtain the two parameters of bubble velocity and void fraction,and better understand the flow ideality of the gas-liquid system.As shown in Fig.11,the ratio of the bubble velocity to the superficial velocity is comprehensively determined by the liquid flow rate,gas injection pressure,and liquid viscosity.Therefore,we select the Weber number of the liquid phase and gas phase to stand the effect of the liquid flow rate and gas flow rate on the dimensionless parameters,respectively,and the capillary number of the liquid phase to represent the liquid viscosity influence.Finally,we obtained the prediction model for the dimensionless bubble velocity as vb/vsup=1.05（WeL/WeG）0.05withR2=0.92,where the applicable range of the value of vb/vsupvaries from 1.0 to 1.7.Fig.13 shows that mathematical models can perfectly predict the bubble velocity within±10%.This model could be applied to the design of gas-liquid microreactors.

Fig.10.(a) Comparison between the experimental data and prediction models in Table 3,and (b) the deviation of the predicted bubble velocity of the models in Table 3.

Fig.11.(a) The effects of the gas injection pressure and liquid flow rate on vb/vsup and α/β,and (b) the effects of the liquid viscosity and liquid flow rate on vb/vsup and α/β.

Fig.12.The distribution of (vb/vsup)·(α/β) at various operation conditions.

Fig.13.Velocity comparison between the experimental data and the predicted values.

To the gas-liquid slug flow in the microchannel,researchers hold the view that this flow pattern could be regarded as the quasi-homogeneous plug flow to some extent,the ratio of the void fraction to the volumetric quality is close to 0.8(see Table 2),thus the ratio of the bubble velocity to the superficial velocity is close to 1.25 according to the mass conservation law.That is to say,the relationship that vb/vsup=1.25 is a universal design criterion to maintain the gas-liquid quasi-homogeneous plug flow state whether the flow regime is the gas-liquid slug flow or the bubble swarm flow.Importantly,the microbubble swarm flow in a relatively large channel can realize the velocity ratio smaller than 1.25,and even reach 1.0,which indicates that the gas-liquid microflow of the bubble swarm is more closely to the ideal plug flow that we desire in the microreactor.Generally,the bubble swarm microflow in a relatively large microchannel has three advantages: (I) the gas-liquid flow is more prone to the idea plug flow,(II) the pressure drop along the channel is smaller than that flow in a relatively small channel,and (III) a relatively large microchannel is beneficial to increase the throughput of the microdevice.

4.Conclusions

In this work,the flow ideality of the bubble swarm is systematically investigated in terms of the void fraction and bubble velocity by the high-speed camera.The results show that the bubble swarm microflow has undergone two conditions: quasi-homogeneous plug flow and pure liquid phase/gas-liquid quasi-homogeneous phase two-phase laminar flow.The void fraction decreases with the liquid flow rate and viscosity but increases with the gas injection pressure,and the absolute deviation of the void fraction first increases with the liquid flow rate to the maximum and then gradually decreases,and the capillary numbers of the transition point are the same when the gas injection pressure is fixed.Besides,the bubble velocity increases with the liquid flow rate and gas injection pressure but decreases with the liquid viscosity,and the absolute deviation of the bubble velocity first increases with the liquid flow rate in a smaller increment rate and then increases in a higher increment rate.Importantly,two transition points of void fraction deviation and bubble velocity deviation simultaneously happen when the ratio of bubble velocity to superficial velocity is close to 1.25,which could be regarded as the design criteria of microchannel critical size.Finally,a new correlation was successfully developed with the Weber number and capillary number for predicting the bubble velocity.This fundamental research of flow ideality analysis could provide valuable general rules for the mass transfer process control,microchannel design,and equipment optimization of microchemical technology.In the future,the modification of the mass transfer coefficient and reaction kinetics based on the actual bubble velocity model will be conducted.Besides,a general bubble velocity model for predicting a swarm of bubbles and a string of bubbles microflow will be tried to develop.

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

We gratefully acknowledge financial support from National Natural Science Foundation of China (21991104).