Longyun Zheng,Kai Guo,*,Hongwei Cai,Bo Zhang,Hui Liu,Chunjiang Liu,2,*

1 School of Chemical Engineering and Technology,Tianjin University,Tianjin 300072,China

2 State Key Laboratory of Chemical Engineering,Tianjin University,Tianjin 300072,China

Keywords:Carbon dioxide Absorption Rayleigh convection Numerical simulation multiple-relaxation-time or generalized lattice Boltzmann model (MRT-LBM)Mass transfer

ABSTRACT CO2 absorption into absorbents is a widely used method to reduce carbon emissions,in which the concentration gradient near the gas-liquid interface may induce Rayleigh convection (RC).Once RC occurs,the mass transfer rate will be significantly enhanced.Therefore,it is necessary to explore the mass transfer enhancement mechanism further and develop a penetration/surface divergence hybrid mass transfer model.In this study,we conduct research on the process of CO2 absorption into ethanol with RC.Firstly,we use a multi-relaxation time lattice Boltzmann method to simulate the absorption process and obtain the flow and concentration fields.And we also verify the reliability of the numerical simulation results by comparing with the experimental results.Then,we analyze the characteristics of non-uniform flow and concentration fields in RC.Moreover,we divide the near-interface region into diffusion-dominated and convection-dominated mass transfer zones by checking whether the horizontal average velocity is greater than 1.0 × 10-4 m·s-1.Furthermore,based on the differences in mass transfer mechanisms of the aforementioned two zones,we propose a penetration/surface divergence hybrid model to predict the instantaneous mass transfer coefficient.The prediction results demonstrate that the hybrid model can precisely predict the instantaneous mass transfer coefficient of the entire CO2 absorption process.Our proposed hybrid model provides a promising way to deal with the complex mass transfer problems with non-uniform flow and concentration fields.

1.Introduction

With the sharp increase in global carbon emissions,the climate warming problem is becoming increasingly serious [1].Carbon dioxide (CO2) capture is an essential and effective way to reduce carbon emissions[2].Generally,the method of CO2absorption into absorbents is commonly used to perform CO2capture in industry.Thus the accurate prediction of CO2absorption rate is greatly rewarding and challenging for the design of absorption equipment and related processes.

In the process of CO2absorption into the stagnant absorbent,the CO2concentration of the liquid beneath the gas-liquid interface increases due to the dissolution of CO2.Consequently,there is a concentration gradient near the interface.The gas-liquid interface becomes unstable as the buoyancy-driven convection structures appear.This convection phenomenon,which is caused by the concentration gradient during the absorption process,is called buoyance-driven Rayleigh convection (RC).As one of the most obvious features of RC,convection structures remarkably enhance the mass transfer rate across the gas-liquid interface[3].The mass transfer enhancement is attributed to the fluid convection that enhances the surface renewal and increases the driving force of mass transfer[4].Therefore,it is essential to study the characteristics of flow and concentration fields in RC and to further explore the mass transfer mechanism behind the complex transport phenomena.

Numerous studies on RC have been conducted by experimental observation,theoretical analysis and numerical simulation to investigate the complex gas-liquid mass transfer mechanism.Experimental observation is well known to be a feasible way to explore the characteristics of RC.The occurrence and evolution of RC have been observed by the schlieren method [3-7] and the pH indicator method[8,9].The characteristics of the flow and concentration fields have been captured by particle image velocimetry(PIV) [10,11] and laser-induced fluorescence (LIF) [12].Moreover,the onset time of RC,which is beneficial to understand the generation mechanism of RC,has been obtained by theoretical analyses,for instance,the critical Rayleigh number analysis [13]and the marginal state analysis [14].It has been verified that the absorption process with RC can be described by nonlinear Navier-Stokes equations and convection-diffusion equations[12].While it is difficult to solve them simultaneously and obtain analytical solutions [15].The lattice Boltzmann method has been becoming more popular in dealing with such complex problems.The lattice Bhatnagar-Gross-Krook (LBGK) model,also usually called the single-relaxation-time lattice Boltzmann model (SRTLBM),has been successfully used to simulate the absorption process with RC [16-18].And the hybrid lattice Boltzmann methodfinite difference method (LBM-FDM) has also been developed[4,10,19-23].In addition,d’Humières [24] developed a multiplerelaxation-time or generalized lattice Boltzmann model (MRTLBM).Compared with SRT-LBM,MRT-LBM had better stability and accuracy as it offered a large number of free parameters[25-30].Recently,MRT-LBM has been continuously developed to improve its applicability and accuracy [15,31].It has been applied to obtain accurate results in the simulation of thermal natural convection [32-37] and thermosolutal natural convection [38-43].However,to the authors’ knowledge,MRT-LBM has rarely been reported to simulate the absorption process with RC.Thus,we attempt to employ MRT-LBM as the numerical method in the simulation of CO2absorption with RC.In addition,in above mentioned experimental and numerical studies on RC,it has been verified that the flow and concentration fields in the liquid layer are spatialvarying and time-varying after the onset of RC.Although these phenomena are complex,there are some notable features.For example,the low-velocity region near the interface will appear with new convection structures.There are high-velocity vortex structures around the convection structure.

Different kinds of mass transfer models were proposed to predict the mass transfer rate of RC.In previous studies,the mass transfer rate was mainly evaluated by incorporating the parameter Rayleigh number(Ra)in empirical correlations[11,44-46].In addition,Huet al.[47,48]proposed a mass transfer coefficient correlation for the CO2absorption in falling film across a wavy interface.They related the mass transfer coefficient to the interfacial vorticity distribution.Guoet al.[20] attempted various turbulent mass transfer models to predict the mass transfer coefficients of CO2absorption in different solvents.Their results indicated that the surface divergence model produced a relatively accurate liquidside mass transfer coefficientkL,t.Zhanget al.[12] predictedkL,tin the convection stable stage based on the penetration model.The contact time of the penetration model was related to the inverse of the average vorticity value near the gas-liquid interface.Geet al.[4,23]proposed a mass transfer model based on a pseudodissipation rate of concentration variance,in which the combined effect of near-surface flow and concentration fields on the mass transfer was considered.

However,it is generally believed that the mass transfer after the occurrence of RC is controlled by convection without thorough consideration of the differences in mass transfer mechanisms of different zones.To fulfill this gap,this study takes the differences in mass transfer mechanisms of different zones into consideration to develop a mass transfer model for good predictions of mass transfer coefficient.For this purpose,firstly,we use MRT-LBM to obtain the instantaneous flow and concentration fields of CO2absorption with RC.Moreover,based on the characteristics of the non-uniform flow and concentration fields,we divide the region beneath the gas-liquid interface into diffusion-dominated and convection-dominated mass transfer zones.Furthermore,according to the differences in mass transfer mechanisms of different zones,we propose a penetration/surface divergence hybrid mass transfer model,where the contribution of diffusion and convection to the mass transfer is fully considered.

The content of this paper is organized as follows:Section 2 presents the governing equations and the penetration/surface divergence hybrid mass transfer model.In Section 3,the numerical scheme of MRT-LBM is introduced,and the accuracy and reliability of the MRT-LBM-based numerical simulation are validated.In Section 4,we discuss the characteristics of flow,concentration fields in different mass transfer stages,and the prediction results of the mass transfer model.Finally,conclusions are summarized in Section 5.

2.Methodology

In this study,as shown in Fig.1(a),the process of CO2absorption into ethanol,which is a typical and widely studied absorption process with RC,is selected as the research object.Within the absorption progress,both the CO2concentration and the density near the gas-liquid interface gradually increase.As a result,the Rayleigh number (Ra) gradually increases.When the maximum value ofRais greater than the critical value,the RC occurs [13].The liquid with a high CO2concentration near the interface moves to the domain far away from the interface.The flow field near the interface becomes complex due to the fluid movement,and the gas-liquid mass transfer is enhanced.In this study,a numerical simulation method was selected to obtain the detailed flow and concentration fields of the occurrence and development of RC.

2.1.Governing equations

The absorption process is simulated within a two-dimensional computation domain.Following the treatment of Refs.[17-23],some assumptions are made as follows.The liquid is regarded as an incompressible Newtonian fluid.The flow is laminar and the viscous dissipation is negligible.The resistance to the gas-liquid mass transfer is mainly in the liquid phase.The thermal effect has been ignored.The Soret and Dufour effect is negligible.The gas-liquid interface is considered to be non-deforming.In the absorption process,the density of ethanol solution increases as the CO2concentration increases.As Δρ/ρ0?1,the density of CO2-ethanol solution is assumed to be linearly correlated with the solute concentration according to the Boussinesq approximation [16],

where ρ0is the density of pure ethanol,andCis the CO2concentration of CO2-ethanol solution.The solutal expansion coefficient βCis calculated as follows,

where Δρ is the density difference between the CO2-saturated ethanol solution and ethanol solvent,andC*is the CO2concentration of the CO2-saturated ethanol solution.

Thus,the body force caused by density difference is approximately given by Shan [16],

wheregis the gravity acceleration.

The absorption process in the liquid phase is controlled by the following two-dimensional macroscopic equations [12].

Fig.1.Schematic diagram of (a) the absorption process and (b) the zone division.

Continuity equation:whereuand v are the velocities in thex-direction andy-direction,respectively.pis the pressure.pcan be written asp=ps+pd.psis the static pressure,ps=ρ0gy,andpdis the dynamic pressure.Thus.

where μ is the liquid viscosity,

Concentration equation:

whereDis the diffusivity coefficient of the solute in the solvent.

The depth of the liquid,H,is taken as the characteristic length[49].Rayleigh number for mass transfer,Ra=H3gΔρ/Dμ,and Schmidt number,Sc=μ/ρ0D,are taken as the dimensionless parameters.

The dimensionless variables are defined as follows [40,41,49].

Then,by substituting the dimensionless parameters and dimensionless variables into Eqs.(4)-(8),the non-dimensional macroscopic equations of the absorption process are derived as follows:

The interface concentrationCItakes the equilibrium concentration[17-21].In a real absorption process,the hydrodynamic instabilities at the interface are mainly caused by momentum,heat,or mass transfer.The generation mechanism of the random instabilities is very complex.For the sake of simplicity,the instabilities are assumed to be induced by interfacial concentration perturbation.A random disturbance model[17]is employed to produce interfacial concentration perturbation randomly.There are two parameters in the random disturbance model:PD,the possibility of concentration perturbation at any given position on the gas-liquid interface,and,CD,the concentration perturbation magnitude.

Thus,the flow and concentration fields can be obtained by solving Eqs.(11)-(14)as well as the random disturbance model simultaneously.The MRT-LBM is chosen as the numerical method to calculate flow and concentration fields.The numerical scheme of MRT-LBM will be shown in Section 3.1.

2.2.Calculation of kL,t

To characterize the interfacial mass transfer,the instantaneous liquid-side mass transfer coefficient,kL,tis calculated as follows:

whereCB,t+ΔtandCB,tare the CO2concentration of the bulk liquid at timet+Δtandt,respectively.CI,t+Δtis the CO2concentration of the gas-liquid interface at timet+Δt,Δtis the time interval,VBis the volume of the liquid,andAIis the area of the gas-liquid interface.

2.3.Penetration/surface divergence hybrid mass transfer model

To accurately predictkL,tin RC,it is necessary to take the influence of hydrodynamic characteristics on the mass transfer into consideration.The mass transfer coefficient predicted by the penetration model is as follows:

where θ is the exposure time.

Previous studies [20,23] have shown that the surface divergence model (SD) [50] exhibited good performance in predicting thekL,tafter the occurrence of RC.The surface divergence model correlates the turbulent mass transfer coefficient with velocity divergence at the gas-liquid interface as follows:

whereScis Schmidt number,and the subscript‘‘I”denotes the areaaveraged value at the gas-liquid interface.

Consequently,kL,tcan be predicted by the surface divergence model as follows:

whereASis the constant of the surface divergence model.

In this study,according to the characteristics of the nonuniform flow field (shown in Fig.1(b)),the overall mass transfer rate is determined by molecular diffusion in the diffusiondominated zone and convective mass transfer in the convectiondominated zone.The instantaneous mass flow rate across the entire gas-liquid interface,Nt,can be calculated as follows:

where the subscripts‘diff’and‘con’denote the diffusion-dominated and convection-dominated mass transfer zones,respectively.

The concentration differences between the gas-liquid interface and bulk liquid in the diffusion-dominated and convectiondominated zones are assumed to be the same,

thus,the overallkL,tcan be obtained by.

The value ofkL-diff,tis predicted by the penetration model from Eq.(16).The exposure time is the resistance time of liquid beneath the gas-liquid interface.The value ofkL-con,tis calculated by the surface divergence model.The hybrid mass transfer model consists of the penetration model and surface divergence model,thus it is referred to as the penetration/surface divergence hybrid model(Pene/SD hybrid model) as follows:

whereASDis the constant of the Pene/SD hybrid model.This is the mass transfer model proposed in this study to predictkL,t,and its prediction results will be shown in Section 4.3.

3.Numerical Simulation Method

3.1.Multi-relaxation time lattice Boltzmann method

In this study,an MRT-LBM with a double distribution model is used to simulate the process of CO2absorption into ethanol.The D2Q9 model is employed,as shown in Fig.2(a).The flow and concentration fields are calculated using distribution functionsfiandgi,respectively.The numerical scheme can be expressed as follows[51]:

wherefandgrepresent the particle distribution functions for the flow and concentration fields,respectively.X is the position vector,ciis the discrete velocity vector,Δtis the lattice time step,fi* (gi*)andfi(gi)represent,respectively,the pre-and post-collision particle distribution functions with discrete velocity ci,M is the transfer matrix,I is a unit matrix,S1and S2are the relaxation matrices offandg,respectively,Fiis the projection of source term at the direction of discrete velocity vector ci.The transfer matrix,M,is as follows [52]:

where ueqis the macroscopic velocity,ωiis the lattice weighting factor.

The values of the weight factors for D2Q9 are:

Fig.2.Schematic diagram of (a) the double distribution D2Q9 model and (b) the boundary conditions.

wherec=Δx/Δtis the lattice propagation speed,the value ofcis set to be 1 [29].Δxand Δtare the lattice grid spacing and time step,respectively.

According to Chopardet al.[54],to improve the accuracy of the LB advection-diffusion model,an external force should be added:

Solvent density,ρ,macroscopic velocity,u,and solute concentration,C,are calculated from the particle distribution functions as follows:

To exert the external force over the flow field,ueqand u should be modified as follows [55,56]:

Here,we select the force scheme following the works of Refs.[17-21].The Rayleigh convection can be well reproduced by the numerical simulation,as shown in Section 3.3.2.To improve the accuracy of the numerical simulation,a rigorous external force scheme should be employed.The detailed introduction to the force scheme can be found in Refs.[31,57].

3.2.Boundary conditions and physical properties

As shown in Fig.2(b),the bottom wall employs the bounce-back boundary for the flow and concentration fields.The left and right walls are periodic boundaries.For the upper boundary of the flow field,a mirror-symmetric boundary is used to treat the free surface(gas-liquid interface).A constant concentration is prescribed on the gas-liquid interface.The conditions of the gas-liquid interface include ?ux/?y=0,uy=0,CI=C*.The initial conditions areC0=0 and u0=0.

The physical properties of pure ethanol and CO2-saturated ethanol solution at 298.15 K and 101.3 kPa are listed in Table 1.ρ0and μ0are the density and viscosity of ethanol solvent,respectively,Δρ is the density difference between the CO2-saturated ethanol solution and ethanol solvent,C* is the CO2saturated concentration,Dis the diffusivity coefficient of CO2in ethanol.

We code Eqs.(23)-(32)along with the boundary conditions and random disturbance model to perform numerical simulations.The simulations are implemented on a HP Z840 Workstation.The two CPUs are both Intel(R) Xeon(R) E5-2697 v3 @ 2.60 GHz.The RAM is 256 GB.The running time is approximately 32 h for Case 1(Grid number,375 × 150,t=0-300 s).

3.3.Validation of numerical simulation

3.3.1.Grid independency test

The grid independence test is conducted.A calculation domain of 25 mm (width) × 10 mm (depth) is discretized separately by using 6250 (125 × 50,Mesh #A),25000 (250 × 100,Mesh #B),56250 (375 × 150,Mesh #C),and 100000 (500 × 200,Mesh #D)square grids.In the test,there is no concentration perturbation at the interface.

The instantaneous mass transfer coefficients obtained from the above four kinds of grid densities are present in Fig.3.The variation trends ofkL,tobtained from different grids are consistent.The time-averaged relative deviation ofkL,tdecreases as the grid number increases.It is merely 1.2%when the grid number is more than 56,250(Mesh#C),but the computational cost increases from 32 h(Mesh#C)to 78 h(Mesh#D).Therefore,grid resolution Mesh#C is adopted in the following simulations.

3.3.2.Validation with experimental results

In this study,we simulate the CO2absorption in the computational domains with different liquid layer heights and widths.The size of the computation domain,grid number,and simulated onset time are shown in Table 2.The onset time of RC is the transition point from molecular diffusion to convective mass transfer,corresponding to a significant increase in mass transfer coefficient.According to Ref.[13],the theoretical onset time of RC for the CO2absorption into ethanol at 298.15 K and 101.3 kPa is 52 s.To precisely reproduce the onset of RC,the values ofPDandCDthat induced the RC at approximately 52 s are adopted in the simulation.According to the calculation results,the concentration perturbation magnitude and the possibility of concentration perturbation in the MRT-LBM-based simulation are,respectively,set to 10-3kg·m-3and 5%.The values ofPDandCDin Cases 1-6 are the same.The temporal-spatial evolution of the convection structures is reproduced in Fig.4.The simulatedkL,tis shown in Fig.5.RC occurs in the initially quiescent liquid layer at 52 s.The size of convection structures gradually increases.Some convection structures approach each other as the liquid moves along with the interface,and finally,they merge into a large-scale convection structure.

Table 3 Time-averaged values of kL,t and RA,con at Stages II and III

Fig.3.Variation of kL,t with time obtained using different grid numbers.

The reliability of the simulation results can be demonstrated from the following aspects.First,the simulated onset time is identical to the theoretical value.Second,as shown in Fig.4(c),there are 15 inverse-mushroom structures in a 70 mm-width liquid layer.In Ref.[4],13 inverse-mushroom structures were observed by schlieren observation within a 60 mm-width liquid layer.Third,as shown in Fig.4(d),at 72 s the front of the convection structures is found to arrive at the liquid layer 20 mm below the interface,which is the same as the experimental result of Ref.[4].Forth,as shown in Fig.5,the peak value of simulatedkL,tis 2.2×10-5m·s-1.After the peak value,the values of simulatedkL,tfluctuated around 1.6 × 10-5m·s-1.Referring to Fig.8 of Ref.[4],the experimental peak value ofkL,tis 2.0 × 10-5m·s-1,and after the peak value,the experimental values ofkL,tfluctuated around 1.6×10-5m·s-1.After the peak value,the values of simulatedkL,tfluctuated around 1.6 × 10-5m·s-1.The deviations in the peakkL,t,and the timeaveraged value within 50 s after the occurrence of RC between the simulation and experiment[3]are merely 5.3%.Thus,the simulation results obtained by MRT-LBM are reliable enough.

3.3.3.Validation with previous numerical simulation results

Guoet al.[20] have simulated the process of CO2absorption into ethanol by using the LBM-FDM method.They also presented the contour of the simulated concentration field and the instantaneous mass transfer coefficient.The two-dimensional simulation domain of their work is 60 mm (width) × 20 mm (depth).The onset time of RC in their work is 52 s,which is consistent with the results of this study.Referring to Fig.3 of Ref.[20],the number,shape,and velocity of the inverse-mushroom structures after the occurrence of RC are similar to the present simulation results,as shown in Fig.4 in this study.Referring to Fig.4(a) of Guoet al.[20],the peak value ofkL,tis 2.7 × 10-5m·s-1,and after the peak value,the experimental values ofkL,tfluctuated around 1.7 × 10-5m·s-1.It indicates that the results in our work are very close to the work of Guoet al.[20].Compared with the simulation results of Guoet al.[20],our simulation results are more consistent with the experimental results [3,4].

4.Results and Discussion

4.1.Characteristics of the absorption process with RC in different stages

The numerical simulation can obtain the liquid layer’s spatialvarying and time-varying flow and concentration fields.Based on it,the characteristics of mass transfer in different stages are obtained.Fig.6 shows the local concentration contours within the liquid layer beneath the interface at different times.

Fig.4.Simulated flow and concentration fields at different times (Case 6).

Fig.5.Comparison of the simulated kL,t and those predicted by the penetration model (Case 6).

As indicated by Zhanget al.[12],the variation ofkL,tcan be divided into three stages.In Stage I,there are no convection structures in the liquid layer(e.g.10 s in Fig.6).The mass transfer is governed by molecular diffusion exclusively andkL,tis well described by the penetration model (e.g.1-52 s in Fig.5).CO2concentrates beneath the interface and the concentration contour becomes thicker in the period of 10-50 s.The liquid velocity is so small that the renewal of the liquid near the gas-liquid interface is not sufficient.Stage II is a convection generation stage,corresponding to 52-80 s in Fig.5.At this stage,as shown in Fig.4(c) and (d),the convection structures appear near the interface,and then gradually extend to the bulk liquid.CO2is transferred from the interface directly to the bulk liquid by the convection structures in contact with the interface,and conversely,fresh liquid,which is also carried by the convection structures,moves up to the region just beneath the interface.As shown in Fig.6,from 52 s to 70 s,the concentration contour beneath the interface is ‘‘squeezed”by the surface renewal.As the RC evolves,the concentration contour around the convection structures becomes thinner,which indicates that the concentration gradient decreases.On the other hand,the horizontal velocity also increases.From the viewpoint of surface renewal,the instantaneous mass transfer is determined by molecular diffusion when the liquid arrives at the interface.The mass transfer rate is inversely proportional to the contact duration between the liquid and interface.The residence time of the liquid on the interface decreases as the horizontal velocity increases.ThuskL,tin Stage II increases owing to the presence of convection structures.In Stage III,kL,tbegins to decrease from 81 s to 120 s,and then it fluctuates within a certain range in the period of 121-300 s.The mass transfer enters a relatively stable state.The number of convection structures decreases as some structures merge into a larger one.The viscous effect gradually dissipates the mechanical energy of the convection structures,and as a result,the convection structures eventually disappear (Fig.4(e)).Meanwhile,new convection structures appear due to the interface’s concentration gradient-induced instabilities.A typical feature of Stage III is the disappearance of old convection structures and the generation of new ones (Fig.4(f)).In Stages II and III,the time-varying and spatial-varying convection structures persist in the liquid layer,which promotes the CO2transfer between the interface and bulk liquid.The penetration model accurately predicts the mass transfer rate in the quiescent liquid layer in Stages I.In Stages II and III,the liquid layer does not remain static,thereforekL,tdeviates significantly from the penetration model.

Fig.6.Concentration contours at different times over the region located 5 mm beneath the interface (Case 6).

4.2.Flow pattern-based zone division near the gas-liquid interface

The precise evaluation of the contribution of different regions to CO2absorption is important for predicting mass transfer in RC.Generally,the near-interface region can be divided into diffusiondominated and convection-dominated mass transfer zones.According to the present results,the average horizontal velocity over the region located 1 mm beneath the interface,ux,ave,can be used to distinguish the diffusion-dominated from convectiondominated zones.The simulation results show that all values ofux,-ave,which are less than 1.0×10-4m·s-1,gradually increase before the occurrence of RC(t＜52 s).Until the occurrence of RC,there are some values ofux,aveexceed 1.0 × 10-4m·s-1due to the acceleration of the vortex on the liquid.It indicates that the value ofux,avemay be closely related to the transition of the mass transfer mechanism.There may be a critical value ofux,avefor the mass transfer transition.Furthermore,we have classified the near-interface region of Cases 1-6 according to different values ofux,avefrom 0.5 × 10-4m·s-1to 1.5 × 10-4m·s-1and obtained the predicted values ofkL,t.Both the classification results of the near-interface region and the prediction performance of the hybrid model show thatux,ave=1.0×10-4m·s-1is a better criterion in this study.Thus,we chooseux,ave=1.0 × 10-4m·s-1as the criterion to divide the diffusion-dominated and convection-dominated regions.For Cases 1-6,the region withux,ave＜1.0 × 10-4m·s-1is considered as the diffusion-dominated mass transfer zone.While the region withux,-ave＞1.0 × 10-4m·s-1is considered as the convection-dominated mass transfer zone.The classification of the near-interface region of Case 1 is illustrated in Fig.7.

Fig.7.Zone division near the gas-liquid interface (Case 1).

Fig.8.Ratios of interface area of convection-dominated zones to the entire interface area of Cases (a) with different layer widths and (b) with different layer depths.

As an obvious feature of the diffusion-dominated zone,the concentration in this zone continuously increases as CO2entered the liquid layer by molecular diffusion.The dense fluid is almost attached to the interface,and cannot be renewed by the bulk liquid.On the contrary,in the convection-dominated zone liquid moves with a larger horizontal velocity,by which the bulk liquid can efficiently renew the liquid.

Fig.9.Comparison of the size of the vortices in (a) Case 6 at different times,(b) Cases with different layer widths,and (c) Cases with different layer depths.

Fig.8 shows the ratios of the gas-liquid interface area of the convection-dominated zones to the entire interface area,RA,con.Before the occurrence of RC,the value ofRA,conis 0,the liquid layer is in a motionless state,and thus the mass transfer is controlled by molecular diffusion.After the occurrence of RC,the gas-liquid interface area of the diffusion-dominated zone rapidly decreases,thus the mass transfer mechanism converts to the combined diffusion and convective mass transfer.As the mass transfer progresses,the diffusion-dominated zone converts to a convection-dominated zone owing to the change of the positions of the convection structures and the new generation of convection structures.On the contrary,the convection-dominated zone converts to a diffusiondominated zone when the convection structures disappear.From Fig.8(a)and (b),it can be found that the area ratioRA,conis deeply affected by the depth of the liquid layer rather than the width.As shown in Table 3,RA,conremains nearly constant when the width of the liquid layer increases from 25 mm to 70 mm,while it increases from 0.54 to 0.83 when the depth of the liquid layer increases from 10 mm to 30 mm.Similarly,the time-averaged value ofkL,tincreases with the liquid layer depth.This may be attributed to the increase in the scale of vortices.

As shown in Fig.9(a),each convection structure consists of two vortices with opposite circulating directions.As the mass transfer progresses,the center of the vortices move downward and the scale of the vortices increases.Att=75 s,several small convection structures merge into a large one,and the convection structure reaches the bottom of the liquid layer.The vortices in the liquid layers with different widths at different times and the ones in the layers with different depths att=75 s are shown in Fig.9(b)and (c).The size of large-scale vortices in the vertical direction is approximately equal to the depth of the liquid layer.Accordingly,both the scale of vortices and the gas-liquid interface area of the convection-dominated zone increase with the liquid layer depth.On the other hand,the horizontal velocity of the vortices in the near-surface region increases approximately by 30 % when the depth of the liquid layer increases from 10 to 30 mm.Thus,the contact duration between the liquid and interface decreases,and the liquid renewal rate around the interface increases.

Fig.9 (continued)

4.3.Regression results of the penetration/surface divergence hybrid mass transfer model

Through Eqs.(18) and (22),the predicted values of the SD model and the Pene/SD hybrid model are obtained.The regression results of the model constants are obtained by fitting the simulated value ofkL,tat Stages II and III.The results are shown in Table 4.

Table 4 Regression results of the Pene/SD hybrid model at Stages II and III

The values ofR2of the SD model are less than 0.55 except for Case 6.As shown in Fig.10(a),there is an obvious difference between the mass transfer coefficients predicted by the SD model and the simulation results at the onset of RC (the beginning of Stage II).This can be attributed to flow inhomogeneity.At the beginning of RC,the number of convection structures is limited,and most of the liquid layer remains quiescent.Thus the contribution of the RC to the mass transfer is not fully represented by using the area-averaged velocity divergence.

The values ofR2of the Pene/SD hybrid model in most of the cases exceed 0.7,which are much larger than those of the SD model.As shown in Fig.10(b),compared with the SD model,the prediction of mass transfer by the Pene/SD model at the beginning of RC improves obviously.This is attributed to the consideration of the differences in the local flow field and mass transfer mechanism in different zones.In addition,the mass transfer coefficients in different regions are precisely evaluated based on the local hydrodynamic parameters.The prediction of mass transfer coefficient by volume-averaged hydrodynamic properties is feasible but not very precise for the mass transfer process with inhomogeneous flow fields like RC.Thus,it is necessary to divide the gas-liquid interface into the diffusion-dominated and convection-dominated masstransfer zones and evaluate the contribution of different zones to the overall mass transfer rate.

Fig.10.Regression results of (a) SD model and (b) Pene/SD hybrid model at Stages II and III (Case 6).

Furthermore,the Pene/SD model is extended to the entire CO2absorption process rather than just limited to the mass transfer process with RC.Table 5 presents the model constants and the values ofR2of the Pene/SD hybrid model for the entire mass transfer process.The values ofR2for the entire CO2absorption process are larger than 0.93.Fig.11 compares the predicted values ofkL,twith the simulation results.It can be found that better agreement between the predicted values and the simulation results comesfrom the precise prediction ofkL,tin Stage I,which is calculated by the penetration model according to the zone division.Overall,the Pene/SD hybrid model predicts well both the variation trend and the values ofkL,tin the entire mass transfer process.As a comparison,the variation trend ofkL,tin Stage I cannot be accurately predicted by the SD model.

Table 5 Regression results of Pene/SD hybrid model at Stages I,II,and III

Fig.11.Regression results of Pene/SD hybrid model at entire CO2 absorption process (Stages I,II,and III) (Case 6).

5.Conclusions

In this study,we have investigated the mass transfer model of CO2absorption into ethanol with RC using MRT-LBM.Based on the numerical simulation results,we have revealed the characteristics of the instantaneous non-uniform flow and concentration fields of the absorption process.Based on the mass transfer mechanism,the hybrid mass transfer model has been proposed and discussed.The following conclusions were obtained as follows.

1.MRT-LBM is a suitable numerical method to simulate the absorption process with RC,which is well verified by the experimental results in the literature.

2.According to the characteristics of the non-uniform flow and concentration fields,the region located 1 mm beneath the gas-liquid interface can be divided into diffusion-dominated zones withux,ave＜ 1.0 × 10-4m·s-1and convectiondominated zones withux,ave＞1.0 × 10-4m·s-1.

3.The results show that our proposed hybrid mass transfer model can precisely predictkL,tof the entire CO2absorption process.

For complex mass transfer problems,the flow pattern-based zone division is beneficial to precisely evaluate the mass transfer mechanism.The proposed hybrid model provides a promising way to facilitate the quantitative study of mass transfer in the complex flow field.We envision that more research will be conducted on CO2chemical absorption with RC in the near future.

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

The authors would like to thank the financial support of the National Natural Science Foundation of China (21706182).