Numerical simulation of local and global mixing/segregation characteristics in a gas–solid fluidized bed

Zhen Wan,Youjun Lu

State Key Laboratory of Multiphase Flow in Power Engineering,Xi’an Jiaotong University,Xi’an 710049,China

Keywords:Discrete element method Solids mixing Binary mixtures Fluidized bed

ABSTRACT Researches on solids mixing and segregation are of great significance for the operation and design of fluidized bed reactors.In this paper,the local and global mixing and segregation characteristics of binary mixtures were investigated in a gas–solid fluidized bed by computational fluid dynamics-discrete element method (CFD-DEM) coupled approach.A methodology based on solids mixing entropy was developed to quantitatively calculate the mixing degree and time of the bed.The mixing curves of global mixing entropy were acquired,and the distribution maps of local mixing entropy and mixing time were also obtained.By comparing different operating conditions,the effects of superficial gas velocity,particle density ratio and size ratio on mixing/segregation behavior were discussed.Results showed that for the partial mixing state,the fluidized bed can be divided into three parts along the bed height:complete segregation area,transition area and stable mixing area.These areas showed different mixing/segregation processes.Increasing gas velocity promoted the local and global mixing of binary mixtures.The increase in particle density ratio and size ratio enlarged the complete segregation area,reduced the mixing degree and increased the mixing time in the stable mixing area.

1.Introduction

Gas-solid fluidized bed reactor is a common industrial reactor.It is widely used in chemical,petroleum,metallurgy and other industries due to its advantages of high combustion efficiency,low pollutant emission and strong fuel adaptability [1–5].In the fluidization process,particles with different physical properties have the tendency to mix or segregate under some operating conditions [6–10].Well-mixing contributes to uniform temperature distribution and efficient mass and heat transfer,which is conducive to fuel combustion and gasification,while segregation is important in fluidized classifiers [11–15].Particles mixing and segregation are one of the remarkable characteristics of fluidized bed technology.A thorough understanding of solids mixing and segregation is necessary for the rational design,improvement,and optimization of fluidized bed reactors.

Many experiments have been previously carried out to study the effect of various operating conditions on particles mixing and separation.Saidiet al.[16] studied air pulsation effect on the segregation efficiency of mixed binary particles in fluidized beds through snapshotting and recording the segregated layer height.They thought the air pulsation was an efficient tool to enhance the segregation process of dissimilar particles.Hartholtet al.[17] investigated the effect of perforated baffles on the mixing and segregation of two-component mixtures and proposed a method to promote the separation of particles in gas–solid fluidized bed.Luet al.[18] studied the fluidization behavior of binary mixtures with different sizes in the bubbling fluidized bed.They analyzed the segregation phenomenon and discussed the relation between the pressure drop curve of the binary mixture and the minimum fluidization velocity.By shutting down the air supply and recording the number of particles,Heet al.[19] investigated the effects of static bed height,gas velocity,and size fraction of iron ore particles on the dry separation efficiency in a dry density-based fluidized bed separator.

To assess the mixing quality many experimental studies choose to freeze the bed by suddenly stopping the gas supply,and divide the bed into a number of sections to evaluate the component fractions of each sample,which can only show the mixing level at the steady state.With the development of computer technology,numerical simulation has gradually become an important method to study the dynamics of multicomponent mixtures,which can show the local transient mixing characteristics.Compared with experments,more information can be easily obtained to investigate fluidization of particles in the fluidized bed.Based on different theories,various numerical simulation methods have been proposed in fluidized beds,such as direct numerical simulation,(coarse-grained) discrete particle method,kinetic method,continuum method and mesoscale-structure-based multiscale method[20].Computational fluid dynamic (CFD) coupling with distinct element method (DEM) can provide detailed micro-scale information,which is suitable for investigating the mixing and segregation phenomenon in fluidized beds [21–23].

Some previous researches were performed to study various behaviors and mixing or segregation patterns of fludized beds by CFD-DEM.Wanget al.[24] studied the mixing characteristics of binary particles in a fluidized bed of refuse derived fuel by CFDDEM.They investigated solid flow characteristics as well as pressure drops and examined the influence of superficial gas velocity on the solid mixing.Penget al.[25] applied a CFD-DEM model to analyze the occurrence of transition from segregation to mixing as the superficial gas velocity increased.They found the vertical fluid force was the main factor responsible for the occurrence.Di Renzoet al.[26] used a coupled DEM-CFD computational code to study the relationship between the mixing equilibrium degree of particles differ in density and the operating velocity and density ratio.A mixing map was provided to characterize the fluidized system containing two kinds of particles.The segregation and mixing of ternary particles in a dense bubbling fluidized bed were investigated to assess four polydisperse drag correlations using CFD-DEM methods by Zhanget al.[27].It was concluded that no model in the study could accurately predict the experimental segregation degree at different gas velocity.Yanget al.[28]evaluated the mixing behavior of a binary mixture with different temperatures in a three-dimensional bubbling fluidized bed based on the coupling of CFD and DEM.The results showed that the time required for a system to reach the temperature dynamic equilibrium was different from that required for the spatial mixing.

Fig.1.Domain of the fluidized bed system.

Some researchers were dedicated to studying the fluidization of non-spherical particles by numerical simulations.Oschmannet al.[29] performed a CFD-DEM simulation of mixing in a model type fluidized bed,and found that mixing as well as bed height progression was strongly influenced by the various elongated particle shapes.The rod-like particles with different aspect ratios were investigated by Maet al.[30],and results showed that particle orientation was controlled by fluidization time,fluidization velocity,and the structure of the fluidized bed.Molaeiet al.[31] employed CFD-DEM approach to study the effects of particle shapes from oblate to prolate on the mixing and segregation phenomenon in liquid fluidization system.Besides,some researches focused on the performance of mixing in different types of fluidized bed by CFD-DEM.Menbari and Hashemnia [32] studied the influence of vibration characteristics on mixing of binary mixtures produced in a vibrationally fluidized bed by using discrete element method,and found that the generation and movement of convection rolls played a fundamental role in mixing.Renet al.[33] carried out numerical simulations for studying the mixing behavior of monocomponent and binary particle systems in a spouted bed.The results showed that spouting gas velocity and particle properties were important to the particle mixing quality in spouted bed.Karimi and Dehkordi [34] used coupled DEM–CFD approach to investigate the mixing process of binary particles in flat-bottom spouted beds.Simultaneous effects of particle specifications,operating conditions and bed dimensions on the equilibrium mixing state of mixtures were evaluated in their work.Other recent researches involving the CFD-DEM approach in the context of mixing or segregation addressed density segregation in dry and wet binary granular mixtures[35],comparisons of Eulerian-Eulerian and CFD-DEM simulations [36] and development of drag models [37].

Most literatures are devoted to studying the global mixing or segregation degree of the entire fluidized bed[38–42].The mixing index used in their works is an overall measure of the mixing stateof the system,which cannot reflect the components distribution of each part of the bed,and thus more detailed local information is required to characterize the real mixing state.Local mixing of solids in a fluidized bed is as an important parameter in the case of rapid reactions where reactants convert to products before bubbles reach the surface or interact with other bubbles in the bed[43].Proper local mixing of solids also prevents the formation of hot spots in the bed [39,44].The local mixing degree and mixing rate are important for understanding the transient and overall mixing process of the bed.In this paper,the CFD-DEM method was employed to investigate the local and global mixing/segregation characteristics in a bubbling fluidized bed.The solids mixing entropy [45] was used to describe the local mixing degree,while averaged mixing entropy was used to calculate the overall mixing level.After verifying the model through experiments,the effects of different factors(superficial gas velocity,particle density ratio and size ratio) on solids mixing or segregation behavior were examined.The global mixing curves and radial-axial maps of mixing entropy and mixing time were obtained.Detailed local information helps to understand the mixing phenomenon and optimize the design and operation of the fluidized bed system.

Fig.2.Global mixing index (S) vs.time (SD+GB,ρj/ρf=1.069, U/Uj=1.802).

2.Mathematical Model and Numerical Method

2.1.Governing equations

The DEM-CFD model treats particles as discrete phase,which are tracked individually by Newton’s second law of motion.The motion of particles can be decomposed into two parts,translation and rotation.The equations of motion for particleiare defined as:

Fig.3.Experimental setup.

wheremiis the mass of particlei;viis the moment of linear velocity;g is the gravitational acceleration;Fdis the fluid drag force;Fcis the contact force;Viis the volume of particlei,andPis the pressure of the fluid phase;Ii,ωi,Tiare the moment of inertia,angular velocity,and the torque of particlei,respectively.In this paper,the contact force Fcis calculated by the widely used soft-sphere approach[46,47].

The fluid phase is considered as a continuous phase whose flow is solved by the local averaged continuity and momentum equations:

Fig.4.Comparison between the experimental and simulated results:(a) Local mixing index (Sb) vs. time at the center of bed;Jetsam concentration profiles (fj) vs.dimensionless bed height (H) in stable fluidization state along the center (b) and wall (c) of bed (SD+GB,ρj/ρf=1.069, U/Uj=1.802).

where ε,ug,ρgare the volume fraction,velocity and density of fluid phase,respectively;Ffpis the volumetric fluid–particle interaction force,and τ is the viscous stress tensor.Drag force is the main force between fluid and particles,thus Ffpcan be calculated as the sum of all the Fdacting on particles in a unit volume,which is generally obtained by multiplying the momentum exchanging coefficient and the slip velocity:

β can be calculated by the conventional Gidaspow drag model[48],which is a combination of Ergun equation [49] and Wen and Yu’s drag model [50]:

whereCdis the drag coefficient,given by:

Reis the Reynolds number of particlei:

2.2.Numerical method

In this work,a quasi-2D fluidized bed with a width of 50 mm,a height of 250 mm and a depth of 0.9 mm was adopted,as shown in Fig.1.Appropriate scaling of the bed is necessary to save computation time,and the bed aspect ratio is a commonly used dimensinless parameter for the scaling of fludized beds.Thus the dimensions of the model were scaled down with the aspect ratio of 1:5 of the experiment.The initial bed height was 100 mm,formed by the free sedimentation of particles.The grid was evenly divided with a size of 2.5 mm × 2.5 mm.Jetsam and flotsam were distributed with a volume ratio of 1:1.The detailed simulation parameters and the properties of particles were listed in Tables 1 and 2.The mixing and segregation processes under different conditions were simulated based on the open source code MFIX [51,52].

Table 2 Properties of particles

Mixing degree is a key factor that affects the operating efficiency of fluidized bed reactors.In order to quantitatively analyze it,scholars have proposed different mixing indexes.The most commonly used index is the Lacey method [53],which is mainly used to evaluate the mean mixing degree of the bed.For the purpose of reflecting the local transient mixing characteristics,the solids mixing entropy [45] was adopted in this study,given by:

whereSbis the solids mixing entropy,which is 0 for the complete segregation state,and the value is 1.0 when the complete mixing state is reached;nis the number of particle species in the bed,which is 2 in this work;αjis the volume fraction of particlejin the statistical area.This index applies to mixtures with the same initial concentration of each solid.Since the volume of the grids is equal,the mixing index of the entire fluidized bed can be defined as the mean value of the local mixing entropy:

Fig.5.Snapshots of the mixing/segregation process vs.time (SD+WS,ρj/ρf=2.103, U/Uj=1.802).

Fig.6.Global mixing index (S) vs. time at three gas velocities.

whereSis the global solids mixing entropy,Nis the number of sampling areas.The emulsion phase and the bubble phase need to be distinguished to obtain the mixing information in the emulsion phase where particles are mainly concentrated.Different threshold voidages of bubble boundary have been used in different researches,such as 0.75 [54],0.8 [55,56],0.85 [57].Boemeret al.[58] thought that the exact value of this definition was not crucial and a voidage of 0.8 was chosen in their simulation.As shown in Fig.2(a),the simulation was conducted when the threshold value was 0.75,0.80 and 0.85.The difference between the results was slight,indicating that the effect of threshold voidage on the mixing index was small,and thus the voidage of 0.8 was chosen as the threshold of bubble boundary in the following simulations.The voidage data in the grid was recorded and outputted,and the sampling area with the voidage of each grid less than the threshold was counted.This required the suitable sampling area size to contain proper particles.The mixing processes for three kinds of sampling area sizes including 2.5 mm × 2.5 mm,5 mm × 5 mm and 10 mm × 10 mm were shown in Fig.2.It could be seen that these sizes had little influence on the mixing trend.In this paper,the sampling area with a size of 5 mm × 5 mm containing four grids was chosen to be used in the following simulations.

Fig.7.Local mixing index (Sb) vs.time at different positions of the bed.The pentagrams represent the experimental result at the measuring point 3 in Fig.1(SD+WS,ρj/ρf=2.103, U/Uj=1.802).

3.Experimental

The experimental setup of a 2-D quasi-fluidized bed is shown in Fig.3.It was equipped with an air compressor,two air valves and rotamers.Evenly distributed air was introduced into the bottom of the bed through a gas distributor.The bed was made of an acrylic sheet with the size of 0.2 m × 0.015 m × 1 m.The circles represented the measuring points,which were distributed along the axial line and near the right wall of the bed.Custom-built capacitance probes were installed at these points to measure the concentration information.Huanget al.[59] proposed a new method to quantitatively measure the mixing and segregation of binary mixtures using capacitance probes,and an expression between the solid volume fraction in the emulsion phase and the probe signal voltage was deduced:

Fig.8.Radial-axial maps of the bed.Initial segregated particle arrangement:(a)final mixing entrophy;(c)mixing time.Initial mixing particle arrangement:(b)final mixing entrophy;(d) mixing time.The black dots represent the value of global final mixing entropy or mixing time examined in Fig.6 (SD+WS,ρj/ρf=2.103, U/Uj=1.802).

Fig.9.Radial-axial maps of the bed.Initial segregated particle arrangement:(a)final mixing entrophy;(c)mixing time.Initial mixing particle arrangement:(b)final mixing entrophy;(d) mixing time.The black dots represent the value of global final mixing entropy or mixing time examined in Fig.6 (SD+WS,ρj/ρf=2.103, U/Uj=1.523).

wheref1is the volume fraction of solid #1 in the emulsion phase,and v is the probe signal value corresponding to the mixture.V1represents the probe signal value when only solid#1 is in the emulsion phase,andV2represents the value when only solid #2 is in the emulsion phase.By recording and processing the probe signal values,the volume fraction of each solid can be gained.Detailed experimental content can be found in our previous work [60].

Geldart’s group B and D particles with different sizes and densities were chosen in the experiment and simulation.Quartz sands and glass beads with the same mean diameter of 582 μm were used in the verification experiment.The lighter glass beads were initially placed on the sands.Fig.4(a) demonstrated the solids mixing entropy versus time at the center of bed in the experiment.Comparing the simulations conducted under the same conditions with the experiment,it could be found that the stable mixing entropy and mixing time were very close.When fluidization entered the stable stage,the concentrations of heavy particles along the axis and wall were obtained.As shown in Fig.4(b) and (c),both the experiment and simulation showed similar distribution of jetsam along the bed height.The model was applicable in this work and more cases were simulated in the following sections.

4.Results and Discussion

4.1.Mixing/segregation process

In the stable fluidization state,the binary particles generally exhibit three states:complete segregation,complete mixing and partial mixing.Each part of the bed shows a similar mixing state when particles are well mixed or segregated.But different mixing levels can be distinguished in the partial mixing state,which deserves more attention and discussion.Typical simulations of mixing and segregation process are presented in Fig.5.Att=0 s,ground walnut shell particles were placed on top of the quartz sands,as shown in Fig.5(a).The injection of air caused the formation and rise of bubbles,which carried the heavy particles to the upper area.With the continuous generation of bubbles,the interaction between gas and solid and the overall circulation of particles were strengthened,promoting the global mixing in the bed.The interface between flotsam and jetsam gradually moved down until a macroscopically stable state was reached at a certain moment.The segregation process shown in Fig.5(b) is the opposite of the mixing process.Ground walnut shell and quartz sands were evenly distributed in the fluidized bed at the initial state.As the bubbles formed and rose,heavy particles sank more rapidly through the space inside the bubbles and gradually accumulated on the bottom of the bed,forming a visible boundary.With the heavy particles increasing in the bottom layer,the boundary moved up until the steady state was reached.After post-processing the particle information through the in-house developed code,we obtained the curves of the global mixing index changing with time,which intuitively reflected the trend of the mixing degree.The blue curves in Fig.6(a) represent the mixing and segregation process of ground walnut shell and quartz sands at 1.802 times the minimum fluidization velocity of jetsam.Although the time it took to stabilize was different,the global mixing entropy tended to a similar degree at the same gas velocity,which was consistent with the phenomenon presented in the snapshots.

For the partial mixing state,the local mixing information is not yet clear in spite of the obvious stratification phenomenon.Thus it is necessary to compare the mixing curves of various positions of the bed.Fig.7 shows the mixing and segregation process of 5 cells on the axis,from which different variations can be identified.The experimental result about the local mixing entropy against time at the measuring point 3 is also added in Fig.7.Due to the continuous generation of bubbles,the volume fraction of jetsam at the measuring point will inevitably undergo huge fluctuations,resulting in a scatter distribution of local mixing entropy.This is caused by the inherent instability of mixing or segregation process in bubbling fluidization,which is consistent with the experimental results.In the mixing process,pure sands basically stayed at the bottom area (cell #1) of the bed.Violent fluctuations of mixing entropy were found in cell #2,indicating that binary particles circulated frequently here.The other three cells in the upper area showed similar upward trends.A simple mathematical equation based on the logarithmic relationship can be used to describe these curves,given by [60]:

whereaandbare two key parameters that need to be recorded and studied.Whent=0 s,Sb=0,representing the initial complete segregation state.After a long period of fluidization,Sbis equal toa,which symbolizes the mixing entropy in the stable state,denoted asSb,sta.The increasing mixing index as a function of time allows the mixing time to be defined by setting a predetermined degree of homogeneity to reach.In this study,the mixing time is defined as the time required to obtain a degree of 99% of the mixing entropy in the equilibrium state(Sb=0.99a),denoted astmix.Thus it is a function of parameterband can be calculated by fitting the data.For the segregation process shown in Fig.7(b),the mixing index in cell #1 quickly dropped to 0,indicating that sands were rapidly deposited at the bottom of the bed.Large changes of mixing entropy versus time were exhibited in cell #2,while the other three cells had only small decreases.These trends are just the opposite of the mixing process at the same position,so a mathematical equation can be applied to fit the data,expressed as:

Fig.10.Global mixing index (S) vs.time for binary mixtures differ in density (U/Uj=1.802).

Fig.11.Radial-axial maps of the bed.Initial segregated particle arrangement:(a)final mixing entrophy;(c)mixing time.Initial mixing particle arrangement:(b)final mixing entrophy;(d) mixing time.The black dots represent the value of global final mixing entropy or mixing time examined in Fig.10 (SD+PP,ρj/ρf=2.88, U/Uj=1.802).

Fig.12.Global mixing index(S)vs.time for binary mixtures(SD+WS,ρj/ρf=2.103,U/Uj=1.802).

The mixing entropy is equal to 1 att=0 s,representing the initial complete mixing state.At the stable fluidization stage,Sbis equal to 1-a.Similarly,tmixis the segregation time,set as the time it takes to reach 99%of 1-a.Since mixing and segregation can be considered simultaneous in the fluidization process,the segregation degree and time are replaced by the mixing degree and time to simplify the expression in the following part.The slopes of the curves changed along the bed height,which meant that the stable mixing entropy and mixing time were different in various positions of the bed for the partial mixing state.Therefore,it is necessary to conduct a local analysis while using the global mixing entropy to describe the mixing/segregation process.In this study,by fitting the dynamic data of all cells to obtain the parametersaandb,the distribution maps of steady mixing entropy and mixing time were drawn and discussed.

4.2.Effect of superficial velocity

The effect of superficial velocity on the global mixing and segregation characteristics of binary mixtures was shown in Fig.6.For the ground walnut shell and quartz sands atU/Uj=1.152,the initially separated particles maintained the segregated state,while the initially well-mixed particles showed a tendency to segregate,and the mixing index gradually decreased from 1 to 0 with time.With the increase of gas velocity,the mixing entropy changed more rapidly and the final degree also increased.The increase in gas velocity accelerated the generation of air bubbles and the overall circulation of particles,which promoted the mixing and shortened the time to reach dynamic equilibrium.The initial arrangement of particles influenced the mixing/segregation process but not the final mixing degree,which was consistent with the results reported in other works [22,25,26].However,this conclusion did not apply to the glass beads and quartz sands shown in Fig.6(b).Although increasing the gas velocity improved the mixing,the initially well-mixed particles did not separate significantly at the three gas velocities due to the small difference in density.At low gas velocities,the final mixing index was not the same in the two cases during the simulation time.In fluidized beds,it is generally believed that bubbles are the main factor to cause particle mixing and segregation.Low gas velocity results in low bubble rising rate and few overall circulation of particles in the axial direction of the bed,so the bed tends to maintain the initial state.For the initially segregated bed at a low gas velocity,bubbles rose through the bed slowly,and only the two groups of particles near the boundary exchanged the position,causing the mixing index to rise slowly during the simulation time as shown by the green solid line in Fig.6(b).Although the bulk density of jetsam decreased due to the expansion of the heavy particles,the volume of the bubbles was small,and the bulk density of jetsam was not sufficiently smaller than that of flotsam,so most particles in the upper layer could not penetrate into the lower layer,which showed a separated state.For the initially well-mixed bed,despite the low gas velocity,the bed was still in a bubbling fluidization regime,and the bulk density of the two particles was always close.Therefore,the binary mixtures were difficult to separate due to the approximate interstitial velocity and gravity,and only a small drop of mixing index was observed during the simulation time as shown by the green dashed line in Fig.6(b).

When the final global mixing entropy of the fluidized bed was 1 or 0,each cell of the bed was correspondingly well mixed or segregated.But for the partial mixing state,the local mixing entropy of each part was not equal to the global mixing entropy.Fig.8(a)showed the final local mixing entropy distribution map of the ground walnut shell and quartz sands atU/Uj=1.802 starting at the complete segregated packing state,and the corresponding mixing time(tmix)distribution map of each cell was shown in Fig.8(c).The black dot represents the value of global final mixing entropy or mixing time examined in Fig.6,which can be identified by the interval of the legend in which it is located.The mixing entropy in the bottom region was 0,and the mixing time was also 0,indicating that pure particles remained at the bottom area of the bed during the fluidization process,which was also consistent with the mixing process shown in Fig.5(a).This area could be referred to as the complete segregation area.In the area further up,the local final mixing entropy showed a distinct increase,and the mixing time also increased significantly to be greater than the global mixing time.These characteristics indicated that the mixing entropy fluctuated sharply,and the binary particles had not reached the stable mixing state in this area,which could be regarded as the transition area.It could be found that position 2 in Fig.7 was in the transition area.Since bubbles tended to form and rise at the center of the fluidized bed,strong particle circulation occurred at the junction,resulting in a long time to reach stability,thus a mixing time gradient from the center to the wall appeared in the transition area.In the upper region of the bed,the distribution of mixing entropy and mixing time was relatively uniform.The local mixing entropy was greater than the global mixing entropy,and the mixing time was close to the global mixing time in this area,which can be called the stable mixing area.The reason was that the mixing entropy in the complete segregation area was less than the global mixing entropy,and the stable mixing area occupied a larger space,where the mixing time could represent the global mixing time of the bed.In addition,the mixing degree along the center line of the stable mixing area was generally slightly larger than at the wall due to the frictional forces near the wall and the tendency of bubbles to pass through the center.It was worth noting that it was better to determine the range of the transition area by the mixing time,rather than the final mixing entropy.Because the final mixing entropy might be close in the three regions,making it difficult to distinguish the transition area,while the mixing time far exceeded the global mixing time in the transition area.

Fig.13.Radial-axial maps of the bed.Initial segregated particle arrangement:(a)final mixing entrophy;(c)mixing time.Initial mixing particle arrangement:(b)final mixing entrophy;(d) mixing time.The black dots represent the value of global final mixing entropy or mixing time examined in Fig.12 (SD+WS, df/dj=1.2,ρj/ρf=2.103, U/Uj=1.802).

Fig.8(b) and (d) showed the distribution maps of local mixing entropy and mixing time starting at the well-mixed state.Likewise,the bed could be divided into the three areas.In the complete segregation area,heavy particles were quickly filtered and deposited at the bottom of the bed.As the bed height increased,the mixing time gradually increased to exceed the global mixing time,entering the transition area.The mixing time was longer at the center of the transition area,and the final mixing entropy showed an upward trend along the bed height.In addition,Sb,stawas zero in the lower part of the transition area,where the steady state was not established yet.The stable mixing area exhibited the similar rule that the local mixing entropy was greater than the global mixing entropy,and the mixing time was close to the global mixing time.Comparing the maps of different particle arrangements,it could be found that the final mixing entropy in the complete segregation area and the stable mixing area were close,but the mixing entropy in the transition area and the mixing time of each part were different.

In order to investigate the effect of superficial velocity on local mixing/segregation process,Fig.9 showed the distribution maps of the local mixing entropy and mixing time of the ground walnut shell and quartz sands atU/Uj=1.523.As the gas velocity decreased,the complete segregation area expanded and more heavy particles accumulated at the bottom of the bed.In the transition area,a part of the final mixing entropy was slightly higher than that in the stable mixing area.This was because the mixing degree in the stable mixing area was small at a low gas velocity,but some particles in the transition area circulated more vigorously,leading to a continuous upward trend of the local mixing entropy.The reduction of gas velocity also reduced the mixing index and increased the mixing time in the stable mixing area.

4.3.Effect of particle density ratio

Fig.10 illustrated the variation in the global mixing entropy with time of binary particles differ in density atU/Uj=1.802.Particle mixtures consisted of spherical particles with a density of 2650 kg?m-3for jetsam and a density of 920,1260 and 2480 kg?m-3for flotsam.It could be seen that decreasing the density ratio increased the slope of the mixing curve,indicating an increase of the global mixing degree and a reduction of the mixing time.For the initially mixed quartz sands and glass beads,no segregation phenomenon occurred at this gas velocity.As the density ratio increased,namely the density of flotsam decreased,the segregation intensified due to the increase of the gravity difference between the two particles.

For the mixtures composed of quartz sands and glass beads,each cell of the bed could quickly reach a uniform mixing state underU/Uj=1.802.The local mixing entropy and mixing time were similar to the global mixing entropy and global mixing time,respectively.But the bed showed a partial mixing state for quartz sands and polypropylene plastics at the same velocity,as demonstrated in Fig.11.Compared with the maps in Fig.8,increasing the density ratio enlarged the complete segregation area and decreased the mixing degree and mixing rate in the stable mixing area.

4.4.Effect of particle size ratio

The effects of particle size ratio on the local and global mixing/segregation characteristics were examined atU/Uj=1.802.Fig.12 showed the trend of global mixing entropy as a function of time for binary mixtures with different sizes.The diameters of ground walnut shell were 0.582 mm,0.698 mm and 0.838 mm,respectively,and the size of quartz sands was 0.582 mm.Compared with the mixing/separation process of binary mixtures with the same particle size,it could be found that increasing the size ratio reduced the final global mixing entropy,and increased the global mixing time.

For the mixtures with a size ratio of 1.2,the global mixing entropy was close to 0.5 in the stable fluidization stage as shown in Fig.12,indicating that each part of the bed underwent different mixing processes.The distribution maps of local mixing entropy and mixing time in this case were presented in Fig.13.As the particle size ratio increased,segregation behavior became more significant.Compared with the cases in Fig.8,more heavy particles were observed to sink to the bottom,causing the mixing entropy in the upper area of bed to drop.In the stable mixing area,the growth of mixing time for the mixing process was slight,but it was huge for the segregation process,which seemed to be related to the magnitude of the variation in mixing entropy.

5.Conclusions

In this study,the CFD-DEM model was applied to investigate the local and global mixing/segregation performance of binary mixtures in a bubbling fluidized bed.A methodology based on solids mixing entropy was applied to calculate the mixing degree and mixing time.Unlike the traditional description of the global mixing degree of the bed,local analysis was highlighted and conducted in this paper.The distribution maps of the local mixing entropy and mixing time were obtained under different operating conditions.Combining the study of global and local characteristics,a more detailed understanding of mixing and segregation process of binary mixtures can be achieved.Main conclusions are summarized as follows.

Each cell of the bed showed a similar degree of mixing in the well mixed or segregated state,while different mixing/segregation processes were demonstrated in various positions in the partial mixing state,where the bed can be divided into three parts along the bed height:complete segregation area,transition area and stable mixing area.For the initial particle segregation arrangement,heavy particles remained at the bottom of the bed.For the initial particle mixing arrangement,heavy particles deposited quickly,and the mixing time increased along the bed height in the complete segregation area.Strong fluctuations of mixing degree were found in the transition area,where the mixing time far exceeded the global mixing time.The mixing entropy in the stable mixing area was greater than the global mixing entropy,and the mixing time was close to the global mixing time.These two initial particle arrangements did not affect the final mixing degree,but influenced the mixing time.Increasing the superficial velocity promoted and accelerated the mixing/segregation process.The increase in particle density ratio and size ratio enlarged the complete segregation area,reduced the mixing entropy and increased the mixing time in the stable mixing area.

Declaration of Competing Interest

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

Acknowledgements

This work was supported by the National Key Research and Development Program of China (2020YFA0714400) and the National Nature Science Foundation of China (51925602,51888103).

Nomenclature

Cddrag coefficient

dfparticle diameter of flotsam,m

djparticle diameter of jetsam,m

dpparticle diameter,m

f1volume fraction of solids#1 in binary solids mixtures

fjvolume fraction of jetsam in binary solids mixtures

Fccontact force,N

Fdfluid drag force,N

Ffpvolumetric fluid–particle interaction force,N

g gravitational acceleration,m?s-2

Hdimensionless bed height

Iimoment of inertia of particlei,kg?m-2

mimass of particlei,kg

Nnumber of sampling area

nnumber of kinds of solids in mixtures

Pfluid pressure,Pa

ReReynolds number

Sglobal solids mixing entropy

Sbsolids mixing entropy

Sb,stasolids mixing entropy in stable state

tmixmixing time spending to reach stable state,s

Titorque of particlei,N?m-1

Usuperficial gas velocity,m?s-1

Ujminimum fluidization velocity of jetsam,m?s-1

Vivolume of particlei,m3

V1voltage signal of solids #1 packing,V

V2voltage signal of solids #2 packing,V

vprobe voltage signal,V

Wdimensionless bed width

αjvolume fraction ofjth solids in mixtures

β fluid–particle inter-coefficient,kg?m-3?s-1

ε voidage

μggas dynamic viscosity,kg?m-1?s-1

ρfdensity of flotsam,kg?m-3

ρggas density,kg?m-3

ρjdensity of jetsam,kg?m-3

τ viscous stress tensor,Pa

vilinear velocity of particlei,m?s-1

ωiangular velocity of particlei,rad?s-1

Subscripts

c contact

d drag

f flotsam

g gas phase

iparticlei

j jetsam

p particle