A route for the study on mass transfer enhancement by adding particles in liquid

Xing Su,Ning Qiao,Bao-Chang Sun,*

1 College of Chemical and Material Engineering,Quzhou University,Quzhou 324000,China

2 State Key Laboratory of Organic-Inorganic Composites,Beijing University of Chemical Technology,Beijing 100029,China

3 Research Center of the Ministry of Education for High Gravity Engineering and Technology,Beijing University of Chemical Technology,Beijing 100029,China

Keywords:Mass transfer Hydrogen Water Methanol Activated carbon

ABSTRACT This work presents a route for the study on the absorption performance of gas into liquid under the condition of adding particles in a stirred constant temperature reactor.Two evaluated systems,hydrogen-water and hydrogen-methanol,with the addition of activated carbon particles (ACP) were carried out,respectively.The results showed that the addition of ACP into the water can enhance the mass transfer between hydrogen and water,the enhancement factor increases rapidly with the increase of the ACP content,and then tends to be unchanged.However,for the hydrogen-methanol system,ACP has little effect on the mass transfer performance.In addition,a gas-liquid mass transfer model considering the effect of solid particle enhancement was established based on the shuttle effect and two-film model.Results indicated that the predicted value agreed well with the experimental value in both hydrogenmethanol-ACP and hydrogen-water-ACP systems.

1.Introduction

Gas-liquid-solid three-phase reactions are very common in the chemical industry,such as fermentation [1,2],hydrogenation [3-6],oxidation,and usually,the gas-liquid mass transfer process is the limiting step for the whole process,therefore,the enhancement of gas-liquid mass transfer is desired for improvement of process efficiency in industry.

Many studies have explored the absorption performance of solid particles in liquid phases.Tingeet al.[7] studied the effect of activated carbon particles (ACP) on the rate of propane and ethane absorption in water in a stirred tank,and found that ACP can increase the gas-liquid mass transfer coefficient up to 3.5 times.Alperet al.[8]studied oxygen absorption in a slurry solution containing ACP,and found that the oxygen absorption rate increased 3 times compared to that without ACP.Ozkanet al.[9]studied the effect of ACP,kieselguhr,Fe2O3and BaSO4on the rate of oxygen absorption in water,respectively,and illustrated that all the particles can significantly enhance the absorption rate of oxygen in the water.However,most of the studies were focused on the intensification performance by adding solid particles,whereas researches on the methodology and exact mechanism of mass transfer near the gas-liquid interface are still rare.

It has been reported that the mechanism of dispersed phase enhancing gas-liquid mass transfer in stirred tanks mainly involve the shuttle effect [8,10] and hydrodynamic effect [11].Demminket al.[12-13]performed C2H4,C2H2,and H2S enhanced absorption experiments with newly generated S particles.Based on the surface renewal theory and the shuttle hydrodynamic effect,a model was established.The experimental results showed that the model can predict the intensification performance by adding ACP and ultrafine TiO2particles into several aqueous solutions[12,14].Brilmanet al.[15] proposed a one-dimensional heterogeneous and non-stationary mass transfer model based on the shuttle effect and penetration model.It was concluded that only the particles being closed to gas-liquid interface can enhance the absorption performance and suggested that two-and three-dimensional models should be developed to better investigate near interface effects.Zhanget al.[16] also developed a general mass transfer enhancement factor model (GEFM) which was based on the shuttle effect and hydrodynamics effect to calculate the overall enhancement factor of mass transfer in the presence of dispersed particles.As mentioned above,modeling for the prediction of absorption enhancement factor has been widely applicated,nevertheless,most of them were usually simplified by using an average mass transfer coefficient instead of the varied one near the gas-liquid interface.Consequently,further theoretical investigations often suffered from lacking of a study approach for the determination of the varied gas mass transfer coefficients in gas-liquid-solid three-phase systems,especially when no experimental data are available.

The aim of this study is to propose a new route for the study on mass transfer enhancement in liquid-containing dispersed particles.Firstly,a device for measuring gas solubility in gas-liquidsolid three-phase systems was designed based on an indirect method by gas phase pressure change.The testing method was verified and a calculation method for gas mass transfer coefficient was then obtained.Secondly,the effect of ACP on the mass transfer of hydrogen in methanol and in water was investigated,respectively.Based on the shuttle effect and two-film model,a threedimensional unsteady heterogeneous mass transfer model for enhancing gas-liquid mass transfer was established and validated by comparison with the experimental data.The effects of operating conditions such as the distance between the particle and the gasliquid interfaceL,the distribution coefficientm,the concentrationmS,and the residence time τ on the enhancement factorEwere analyzed.

2.Experimental

2.1.Experimental procedure

The experiment setup for measuring gas solubility and liquid volume mass transfer coefficients in liquid is schematically shown in Fig.1.It mainly consists of a hydrogen cylinder,equilibrium chamber,gas chamber,thermostatic water bath and magnetic stirrer,and vacuum pump.The gas chamber with a volume of 273 ml was used as a buffer tank and put in the water bath for preheating in advance.The magnetic stirring devices were arranged under the equilibrium chamber and the gas chamber,respectively.Pressure transmitters(Keller 33x,Switzerland;0-10 MPa,0.05%)were connected to the upper part of the equilibrium chamber and the gas chamber,which can record the pressure in the two chambers in real time.The temperature of the equilibrium chamber and the gas chamber were both controlled by the water bath,and the temperature control accuracy of the water bath is±0.05°C.Gas-liquid mass transfer occurred in the equilibrium chamber,and its volume is 140 ml.

Additionally,in order to exclude the influence of temperature change on gas pressure,the following requirements were made for the measuring device.(a) The accuracy of temperature control should be within ±0.1 °C.According to the ideal gas equation of state (at 1 MPa and 30 °C),when temperature changes by 1 °C,the gas pressure will change by 3 kPa.In this paper,we want to study the effect of particles on H2solubility and liquid volume mass transfer coefficient,and this effect could be very small;(b)the gas should be sufficiently preheated to the experimental temperature in the gas chamber 4.

To test the reliability of the device,the solubility of H2in toluene was firstly tested with the device at 25 °C and the results were compared with literature data [17],as shown in Table 1.It was found that the maximum error was only 1.1%.

The experimental process can be briefly described as follows:Evacuated the entire system,and closed all valves after evacuating.The degassed solution entered the equilibrium chamber through a buret from valve 6.The amount of solution entering the equilibrium chamber was obtained by reading before and after the buret,and the liquid temperature in the buret was measured at the same time to estimate the change of the amount of solution.Filled the gas chamber with hydrogen from the hydrogen cylinder.Turned on the heating and stirring,and kept the system stable for 1 h after reaching the set temperature.Stopped stirring,at this time,the pressure of the equilibrium chamber was recorded asp0.Opened the valve 8 to slowly fill the equilibrium chamber with hydrogen from the gas chamber,and closed the valve 8 after the pressure of the equilibrium chamber reachedp1.Turned on the agitation and recorded the pressure changes over time.When the pressure was not changing,recorded the pressure in the equilibrium chamber,the pressure in the equilibrium chamber was recorded asp2.

The ACP used in the experiment was purchased from Xianfeng Company (Nanjing,China),and its main properties were listed in Table 2.Hydrogen was purchased from Haipu Company (Beijing,China) with a purity of 99.999%.

Table 1 Comparison of experimental value and literature value

Table 2 Properties of used ACP

Table 3 Simulation parameters

2.2.Determination of liquid volume mass transfer coefficient

Based on the parameters obtained from experiments mentioned above,the method for determination of liquid volume mass transfer coefficient is described as follows:

The mass transfer coefficient of the liquid phase was measured by the gas intermittent absorption technique.The principle of the gas intermittent absorption technique is to transform the measurement of liquid phase concentration into the measurement of gas phase pressure according to the classic two-film theory.When deriving the expression of liquid volume mass transfer coefficient,the basic assumptions were made as follows:(1)the absorption of gas in liquid can be described by Henry’s law,due to the relatively low hydrogen solubility in liquid at low pressure;(2) when filling the equilibrium chamber with hydrogen from the gas chamber,the gas absorption in liquid before the start of stirring can be neglected;(3) the gas phase and liquid phase temperatures in the equilibrium chamber can be considered to be uniform and constant,because the changes of gas pressure and liquid concentration were both extremely minor during the experimental process;(4)mass transfer resistance in gas phase was ignored due to only pure gas was used;(5) the liquid phase can be considered to mix uniformly in the equilibrium chamber and have a uniform concentration at any time when the rotation rate was greater than or equal to 500 r·min-1according to previous experiments;(6)phase interface instantly reached the phase equilibrium.According to the above assumptions,the derivation process of the liquid volume mass transfer coefficient was as follows:

The absorption rate of gas is equal to the rate at which the gas phase pressure decreases:

According to the two-film theory,the differential mass equilibrium equation of the liquid phase is as follows:

Fig.1.Schematic of the experimental setup: (1) hydrogen cylinder,(2) thermostatic water bath and magnetic stirrer,(3) equilibrium chamber,(4) gas chamber,(5,6) pressure transmitter,(7) computer,(8-11) valve.

According to Henry’s law,the solubility of gas at low pressure is directly proportional to the partial pressure of the gas.The Henry constant is only related to temperature.

According to assumption (6),the concentration at gas-liquid interface is equal to:

Then the concentration in liquid bulk can be calculated by:

Substituting Eqs.(3) and (5) to Eq.(2) can obtain the following equation:

The boundary conditions of Eq.(1) are:t=0,p=pi,NL=0,then the following integral equation can be obtained,wherepiis the initial pressure in the equilibrium chamber after inflation of gas.

Substituting Eqs.(1) and (7) into Eq.(6),we can get:

Considering the vapor pressure of the solvent,the final formula is:

If the left item of Eq.(10)is recorded as function,liquid volume mass transfer coefficientkLacan be calculated by the linear slope of the functionvs tplot.

Typical pressure changes with time were shown in Fig.2.And the function-tplot was shown in Fig.3.According to the value of slope in the fitting curve formula in Fig.3,the liquid volume mass transfer coefficientkLacan be obtained as 0.003301.

3.Model

In order to describe the effect of solid particles on the gas-liquid mass transfer process,different mass transfer enhancement models have been established,which mainly combined the classical gas-liquid mass transfer theory (two-film theory,permeability theory and surface renewal theory)with the mass transfer mechanism of gas adsorption in liquid in presence of solid particles(shuttle effect and hydrodynamic effect).The shuttle effect is that the dispersed phase can move into the liquid film under the entrainment of the fluid or Brownian motion,and stay in the film for a period of time.After adsorbing a certain amount of component to be transferred,the dispersed phase returns to the liquid bulk.The particles are regenerated after desorption and a transport process is completed.The hydrodynamic effect is that the dispersed particles in the multiphase system affect the hydrodynamics behavior of the system by colliding with the gas-liquid interface or inducing turbulence near the gas-liquid interface,resulting in the decrease of the diffusion layer thickness,thus enhancing the mass transfer.

On the basis of the shuttle effect and two-film model,a model of mass transfer enhancement by solid particles was established.The mass-transfer mechanism was shown in Fig.4.First,the particles with adsorption ability move to the gas-liquid interface,and absorb gas components,then are transported by fluid to the liquid bulk to desorb,and this process cycles back and forth to achieve mass transfer enhancement.Based on the theories of hydrodynamics and mass transfer,the following assumptions were made:

(1) The concentration of dispersed particles in the gas-liquid interface is different from that in the bulk of the liquid phase.Langmuir isotherm adsorption equation can be used to describe the relationship between the two concentrations [18].

Φsand Φs,maxare the volume fraction of the dispersed particles in the gas-liquid interface and the maximum volume fraction in the liquid phase,respectively.Andkais the adhesion coefficient.Φs,maxandkaare affected by stirring speed,particles size and wettability of particles.

Fig.2.Graph of pressure over time (H2,500 r·min-1,30 °C).

Fig.3.Function-t graph (H2,500 r·min-1,30 °C).

Fig.4.Schematic diagram of mass transfer route.

(2) The theory of turbulence indicates that the contact time of gas and liquid can be estimated by the time scale of turbulent vortices[19].Therefore,the residence time of the particles in the gasliquid interface can be considered to be equal to the gas-liquid contact time.

(3)The particles size in liquid phase is uniform,and the average particle size is adopted in this work.

The computational domain of this system was shown in Fig.5.According to Nagyet al.[20],the distanceLbetween the particles and the gas-liquid interface can vary between 0 and δx,where δxis the distance between adjacent particles which can be determined by δx=d(1/-1),where Φsis the particles volume fraction,dis the particles diameter.In these simulations,Lwas chosen as δx/2 and the values of δx,δyand δzwere all equal to δx+d.

For continuous phase,the following equation can be obtained:

Fig.5.Diagram of the model.

wherecAis the concentration of the component to be transferred in the continuous phase,DAis the diffusion coefficient of the component in the liquid phase,which can be calculated byDA=D0(μ0/μ),andD0is the diffusion coefficient of the components in the blank base liquid,μ0is the viscosity of blank base liquid,μ is the viscosity of continuous phase,which is calculated by μ=μ0(1+2.5Φs) [21].

For the dispersed phase,the differential equation ofcA,dcan be described as follows:

Among them,cA,dis the concentration of the component in the dispersed phase,DA,dis the diffusion coefficient of the component in the dispersed phase.

The initial conditions for Eq.(14) are:

wherec* is the concentration of the component in the gas-liquid interface,which can be obtained by Henry’s Law:c*=p/H,pis the absolute pressure of the gas,andHis Henry’s coefficient.

The relationship between the boundary conditions on the interface of continuous phase and dispersed phase are as following:

Among them,mis the distribution coefficient,indicating the adsorption capacity of dispersed particles.

Complex mass transfer differential equations and boundary conditions can be solved by numerical methods.Using the finite element method,we can get the gas concentration at any time and any position of the liquid in the system by the software of COMSOL Multiphysics 5.4.When the concentration distribution is obtained,the mass transfer fluxJin the presence of nanoparticles can be obtained by the following formula:

In the absence of particles,the mass transfer flux can be obtained by the following formula:

Then,the enhancement factor can be expressed as:

Fig.6.Mass transfer coefficient of hydrogen absorption in water and the enhancement factor at different content of ACP (30 °C,1 MPa,500 r·min-1).

4.Results and Analysis

4.1.Effect of solid content on mass transfer coefficient and enhancement factor

The influence of ACP content on the mass transfer coefficient and enhancement factor of hydrogen absorption in water and water-ACP system were shown in Fig.6.It was illustrated that both the mass transfer coefficient and enhancement factor increase rapidly to about three higher times with the content of ACP increasing from 0 to 0.02 kg·m-3,and then tends to level.The improved mass transfer performance can be mainly attributed to the gas adsorption by ACP and the shuttle effect.On the one hand,the hydrogen absorption capability of ACP is much greater than the solubility of hydrogen in water according to the very high distribution coefficient of hydrogen in water with ACP,which was reported as 140 by Dagaonkaret al.[22].On the other hand,the ACP could move to the gas-liquid interface to absorb hydrogen,and then return to the liquid bulk to release hydrogen (the shuttle effect).Thus,due to the ACP constantly moved back and forth between the liquid film and the liquid bulk,resulting in the significantly enhanced gas-liquid mass transfer.However,because the enhanced mass transfer was also limited by the ACP content on the gas-liquid interface,which was not equal to that in the main body of liquid phrase,thus,when the ACP content increased to a certain value in the liquid bulk,the ACP content on the bubble surface reached its saturated amount,leading to the enhanced mass transfer performance became unchanged with the content of ACP higher than 0.02 kg·m-3.

The influence of ACP content on the mass transfer coefficient and enhancement factor of hydrogen in methanol and methanol-ACP system were both shown in Fig.7.It was demonstrated that the ACP have little effect on the mass transfer coefficient of hydrogen in methanol.Due to the distribution coefficient of hydrogen in methanol with ACP is as low as 22,obtained by Eq.(17),the effect of ACP on the solubility of hydrogen in methanol can be neglected,thus,the enhancing ability was limited.Besides,the hydrophobic property of the liquid can also influence the mass transfer coefficient.Wimmerset al.[23] studied the concentration distribution of ACP in ethanol aqueous solution at the gas-liquid interface as well as the main body of liquid phase.The adhesion ability of ACP on the bubble surface was found to be decreased with the increase of ethanol content in the ethanol aqueous solution.Therefore,for the ACP-methanol system,it can be deduced that the ACP in pure methanol is not as easily absorbed into the gas-liquid interface as that in water,because the properties of methanol are similar with ethanol rather than water,thus the ACP content near the gas-liquid interface could be lower in the ACP-methanol system,resulting in no promoting effect on the mass transfer of H2in methanol.

Fig.7.Mass transfer coefficient of hydrogen absorption in methanol and the enhancement factor at different content of ACP (30 °C,1 MPa,500 r·min-1).

4.2.Comparison of experimental data and model

The parameters needed for mathematical model calculation were listed in Table 3.

Figs.8 and 9 were the comparison of experimental results and model calculations in ACP-water system as well as in ACP-methanol system.It was revealed that the trends of experimental data were consistent well with the calculation results of the model in both systems.However,in Fig.8,certain deviations between calculation and experiments occurred when the content of ACP increases to 0.02 and 0.03 kg·m-3,which might be due to the errors during the experiment of measuring.All the experimental data were used to calculate the total standard deviation (SD) between the calculation and experiments by using the formula of Eq.(21),and the result turned out to be only 7.78%.In addition,it was shown in Fig.10 that the deviations mainly came from the ACPwater system rather than ACP-methanol system,where the four data point in ACP-methanol system were happen to overlap so that they looked like one data point.

4.3.Modeling analysis of ACP enhanced gas-liquid mass transfer

According to the above description of gas-liquid mass transfer process in the presence of particles,the main parameters affecting the enhancement factor were the distance between particles and gas-liquid interfaceL,distribution coefficientm,the content of particlesms,and the residence time of the particles in the gas-liquid interface τ,here the τ can be considered to be equal to the gasliquid contact time.It is interesting to study the effect of these four parameters on the enhancement factor by using the model established in Section 3.

Fig.8.Comparison of experiment and model for hydrogen absorption in ACP-water system.

Fig.9.Comparison of experiment and model for hydrogen absorption in ACPmethanol system.

Fig.10.The comparison of the experiments and calculation.

Fig.11.Effects of the distance L between the particle and gas-liquid interface on the enhancement factor (DA=2 × 10-9 m2·s-1, DA,d=4 × 10-10 m2·s-1, d=5 μm,m=100, ms=0.35% (mass)).

Lis one of the most important factors influencing the enhancement factor of gas-liquid mass transfer.Fig.11 showed the effect ofLon the enhancement factor by model calculating with the variation ofLfrom 0.5dto 2d.A small value of theLmeans a short distance between ACP in liquid and gas phase,and the diffusion resistance can be greatly reduced,which are beneficial to the mass transfer from gas to liquid phase.Therefore,the smaller theL,the greater the enhancement factor as shown in Fig.11.It also proved that the adsorption process of ACP is the limiting step when the liquid is in the turbulent state.When theLdecreases to a certain value,the diffusion resistance between ACP and gas can be neglected,corresponding to the sharp increase of enhancement factor at the beginning.On the contrary,when the diffusion distanceLis large,it takes a long time for the gas diffusing to contact the ACP,and thus,the enhancement factor increases more slowly with the increase of the residence time.In Fig.11,it was noted that whenLdecreases to 0.5d,the enhancement factor represents the most drastically increasing trend with reference time increase at the beginning,in which the maximum enhancement factor up to 2.5 could be obtained.WhenLis twice the ACP diameterd,the enhancement factor represents the most slightly increasing trend,in which the maximum enhancement factor is only about 1.1.Moreover,the residence times corresponding to the maximum enhancement factor were shown to increase with the increase ofLin Fig.11.For example,it was observed as 0.05 s atLof 0.5dand 0.12 s atLof 2d,respectively,this is because the time required for ACP to reach saturation becomes longer due to the slower process for the gas diffusing to contact the ACP as discussed above.

The distribution coefficientmrefers to the solute partitioning between the solid particles and the liquid phase [12],which reflects the adsorption capacity of the particles in liquid.A larger distribution coefficient usually means a greater adsorption capability for solute by particles,and consequently,a longer time is needed to achieve the saturated state of the particles.Fig.12 showed the effect of different distribution coefficient on the enhancement factor by varying themfrom 1 to 200.It was illustrated that a larger distribution coefficient results in a higher enhancement factor.Additionally,it should be pointed out that,in Fig.11,and Fig.12,there are some minor differences between the starting values,which are caused by the rapid changes at the beginning of the residence time under different operating conditions based on the model calculation.

Fig.13 showed the effect of particle concentration on the enhancement factor.It was demonstrated that the increasing rate of enhancement factor is more rapid with a high content of ACP at the beginning of the reference time.It was noted that with the increase of ACP content,the residence time required to reach the maximum enhancement factor becomes shortened.This is because the model is a single particle model,which means a larger ACP concentration leads to a smaller calculation area.With increasing the content of ACP,Lbecomes small,and thus,the saturation state is easier to be reached.

Fig.12.Effects of the distribution coefficient on the enhancement factor(DA=2 × 10-9 m2·s-1, DA,d=4 × 10-10 m2·s-1, d=5 μm, L= d, ms=0.35% (mass)).

Fig.13.Effects of particle mass concentration on the enhancement factor(DA=2 × 10-9 m2·s-1, DA,d=4 × 10-10 m2·s-1, d=5 μm, L=δx/2, m=100).

It was also observed from Figs.11-13 that all the enhancement factors increase up to a maximum and then decrease with the increase of residence time.This phenomenon can be explained by the following point of view.Before the ACP reach the saturated state,the absorption capacity of the ACP is higher than that of the solvent,presented as the enhancement of mass transfer.After the ACP reach the saturated state,the ACP can no longer adsorb the components,and thus the enhancement effect disappears gradually.Therefore,after the residence time exceeds the time when the ACP reach the saturated state,the enhancement factor will decrease as the time continues.

5.Conclusions

In this paper,we proposed a route of study on the absorption performance of gas into liquid containing solid particles in a stirred constant temperature reactor.A measuring device for gas solubility was built,and a calculation method for the liquid volume mass transfer coefficient was established.It was revealed that ACP can increase the mass transfer coefficient by about three times in the ACP-water system,while in the ACP-methanol system,this effect turned out to be very little and can be ignored.With the increase of ACP content in water,both the mass transfer coefficient and the enhancement factor increase rapidly up to a maximum,and then tends to level.Based on the shuttle effect and two-film theory,a three-dimensional unsteady heterogeneous mass transfer model was also established.The effects of the distance between the particle and the gas-liquid interfaceL,the distribution coefficientm,the content of ACPms,and the residence time τ on the enhancement factorEwere all analyzed by modeling calculation.It was found thatEdecreases with the increase ofL,whenLis more than 2 times the particle diameter,the enhancement factor is close to 1,and the ACP could hardly enhance the gas-liquid mass transfer coefficient;the enhancement factor increases with the increase ofm,and when themis less than 10,the enhancement of the ACP to the mass transfer coefficient can be ignored;the enhancement factor increases as the content of ACP increase;Eincreases first and then decreases as the residence time increases.The total standard deviation (SD) was shown to be only 7.78%,indicating a good agreement between the calculation and experimental results.

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 National Key Research and Development Program of China(2019YFA0210302),the National Natural Science Foundation of China (21878009,21901141),and Basic Research Plan for Public Welfare of Zhejiang Province,China(LZY21B060001).

Nomenclature

c* liquid phase concentration in equilibrium with gas pressure,mol·m-3

cAconcentration of components to be transferred in the continuous phase,mol·m-3

cA,dconcentration of components to be transferred in the dispersed phase,mol·m-3

cLconcentration of components to be transferred in the liquid bulk,mol·m-3

cLiconcentration of components to be transferred at the gasliquid interface,mol·m-3

DAdiffusion coefficient in continuous phase,m2·s-1

DA,ddiffusion coefficient in dispersed phase,m2·s-1

D0diffusion coefficient in blank base solution,m2·s-1

dparticles diameter,m

Eenhancement factor

kaparticles adhesion coefficient,m3·kg-1

kLaliquid volume mass transfer coefficient,s-1

mdistribution coefficient

mscontent of particles in suspension,kg·m-3

NGamount of components to be transferred in the gas phase,mol

NLamount of components to be transferred in the liquid phase,mol

pfpartial pressure of hydrogen in the equilibrium chamber after equilibration,Pa

pipartial pressure of hydrogen in the equilibrium chamber after inflation,Pa

p0pressure in the equilibrium chamber before inflation,Pa

p1pressure in the equilibrium chamber after inflation,Pa

p2pressure in the equilibrium chamber after equilibration,Pa

Rideal gas constant

Ttemperature,K

VGvolume of the gas phase in the equilibrium chamber,m3

VLvolume of the liquid phase in the equilibrium chamber,m3

ε turbulent energy dissipation rate,W·kg-1

μ dynamic viscosity,Pa·s

τ residence time,s

υ kinematic viscosity,m2·s-1

Φsvolume fraction of dispersed phase at gas-liquid interface

Φs,maxmaximum volume fraction of dispersed phase at gas-liquid interface