Analysis of the flow pattern and periodicity of gas-liquid-liquid three-phase flow in a countercurrent mixer-settler

Minghang Zhang,Wangfeng Cai,Pei Zhu,Le Xie,Yan Wang,*

1 School of Chemical Engineering and Technology,Tianjin University,Tianjin 300350,China

2 College of Chemistry and Chemical Engineering,Central South University,Changsha 410083,China

Keywords:Countercurrent mixer-settler Three-phase flow Sliding mesh model (SMM)Fast Fourier transform (FFT)

ABSTRACT In contrast to the concurrent mixer-settler,the interaction between the mixing and settling chambers have to be taken into account in the simulation of the countercurrent mixer-settler,and no work has been reported for this equipment.In this work,a three-phase flow model based on the Eulerian multiphase model,coupled with a sliding mesh model is proposed for a countercurrent mixer-settler.Based on this,the dispersed phase distribution,flow pattern,and pressure distribution are investigated,which can help to fill the gap in the operation mechanism.In addition,the velocity vector distribution at the phase port shows an intriguing phenomenon that two types of vectors with opposite directions are distributed on the left and right sides of the same plane,which indicates that the material exchange in the mixing and settling chambers is simultaneous.Analysis of this variation at this location by a fast Fourier transform(FFT)method reveals that it is mainly influenced by the mixing chamber and is consistent with the main period of the outlet flow fluctuations.Therefore,by monitoring the fluctuation of the outlet flow and then analyzing it by the FFT method,the state of the whole tank can be determined,which makes it promising for the design of control systems for countercurrent mixer-settlers.

1.Introduction

As a kind of extraction equipment,the mixer-settlers are widely used in rare earth separation[1],wet metallurgy[2],and spent fuel treatment [3].Up to now,many different types of mixer-settlers have been developed for extraction separation due to the advantages of simple structure and easy operation when compared with rotating disk contactors and pulse columns,etc.[4-6].Mixersettlers usually have two operation modes: concurrent and countercurrent,depending on the flow direction of organic and aqueous phases,as shown in Fig.1.Although they both contain mixing chamber and settling chamber,their hydrodynamic characteristics and mass transfer effect are significantly different.

Over the past several decades,mixer-settlers have received a lot of attention due to their wide application.Many efforts have been devoted to optimizing the structural parameters(i.e.,impeller type and baffle) [7,8] and exploring the dispersion behavior of the droplets [9,10].Additionally,computational fluid dynamics (CFD)modeling and simulation have been used in recent years as a visualization tool for the study of mixer-settlers.Yeet al.[11,12]studied the flow pattern in the settling chamber using a combination of particle image velocimetry experiments and CFD simulations,and optimized the baffle.Tanget al.[13] studied the droplet size distribution in the mixing chamber using a population balance model.As can be seen,the mixing and settling chambers are usually investigated separately due to the complex flow patterns.However,it is only applicable to the concurrent mixersettler,but not to the countercurrent mixer-settler.Generally,the fluid flow at the connection between the mixing and settling chambers in a concurrent mixer-settler is unidirectional,but the situation is different for a countercurrent mixer-settler,where the connection acts as both an outlet and an inlet.As a result,the simulation of the countercurrent mixer-settler must analyze its mixing and settling chambers together.This makes the simulation more complicated because the flow states of the mixing and settling chambers are completely different,and very little has been reported in this regard.

In the counter-current mixer-settler,the ratio of aqueous and organic phases is different in the different zones.Therefore,in order to make the pressure balance,there is inevitably a liquid level difference between adjacent chambers.Moreover,during the operation of the equipment,the liquid level will also change with time.Therefore,air must also be taken into account in the simulation process to reflect the liquid level variation.Chenget al.[14]used the Eulerian multiphase model to numerically simulate the gas-liquid-liquid system in the stirred tank to study its macroscopic mixing process.Liet al.[15] used the volume-offluid multiphase model to investigate the hydrodynamics characteristics of a gas-liquid-liquid three-phase system inside an annular centrifugal contactor.Therefore,for rotating devices,the Eulerian model is a promising model for multiphase flow.

The flow pattern in mixing equipment is not invariant when reaching stability,but alternately changes from one type to the other.This situation is called macro-instability(MI).Many scholars have found that MI has a significant effect on the mixing performance in multiphase systems [16-18].For countercurrent mixersettlers,it is uncertain whether the flow pattern differs from that of a regular mixing tank due to the presence of the phase port,so studying its MI helps to understand its flow mixing state.

In addition,although countercurrent mixer-settlers have been used in industry for many years,the flow pattern at the phase port is very complex and the details are still unknown.Therefore,it is particularly important to analyze its time-varying characteristics.Fourier transform [19],as a signal processing method,is able to convert the time signal into a frequency signal to derive the periodic pattern of the parameter variation.The application of the Fourier transform in industry lies mainly in control systems [20-22],but in recent years it has also been used by some researchers in CFD simulations.The National Energy Technology Laboratory uses the fast Fourier transform (FFT) to quantify turbulent flow in gas-liquid bubble columns [23-26].Hoqueet al.[27] evaluated the pressure spectrum of a two-phase flow in an oscillating grid system using the FFT method.Janigaet al.[28] characterized the MI of the stirred tank using FFT analyses of the threedimensional proper orthogonal decomposition temporal coefficients.However,there are no reports of FFT analysis for mixersettlers.

In the present work,Eulerian three-phase flow model coupled with RNGk-ε model was developed to investigate the fluid flow behaviors in the mixing chamber and settling chamber simultaneously for a countercurrent mixer-settler.The sliding mesh model(SMM) was used to simulate the rotation of the impeller.The dispersed phase distribution,pressure distribution and velocity vector distribution were investigated throughout the tank,which help to understand the fluid flow mechanism in the countercurrent mixersettler.Since the flow pattern near the phase port between the mixing and settling chambers is extremely complex and important,the fast Fourier transform method was employed to describe the time-varying nature of its associated variables,and the causes of the periodic variation are subsequently obtained by analyzing the velocity vector.

2.Physical Model

2.1.Process description

Fig.2 illustrates the mixing and separation process of the aqueous and organic phases in a counter-current mixer-settler,with the gas phase located above the liquid phase,which plays the role of controlling the liquid level.The flow route of the aqueous phase is: (i) water is pumped from the storage tank into the left settling chamber and crosses over the left overflow weir into the mixing chamber.(ii) It is mixed with the organic phase in the mixing chamber and then crosses the right overflow weir into the dispersion band.(iii)After the water passes through the dispersion band and settles,the liquid level on the right side of the settling chamber rises.The water is discharged from the right settling chamber.The flow route of the organic phase is: (i) oil is pumped from the storage tank into the right settling chamber and crosses the right overflow weir into the mixing chamber.(ii)In the mixing chamber,it is mixed with aqueous phase and then enters the dispersion band through the phase port.(iii) Due to the density difference,the oil buoys from the dispersion band into the organic phase in the left settling chamber,causing the liquid level to rise.Finally the oil is discharged from the left settling chamber.

In this study,deionized water was used as the aqueous phase,kerosene as the organic phase,and air as the gas phase.The main physical properties,including density,viscosity,and surface tension coefficient,are shown in Table 1.

Table 1Physical properties of chemicals used at 20 °C

2.2.Physical modeling and meshing

In this study,a countercurrent mixer-settler with dimensions of 450 mm × 120 mm × 200 mm was used,as shown in Fig.3(a).It consists of three chambers,with the settling chamber on both sides and the mixing chamber in the middle,separated by baffles.In the mixing chamber,a coaxial double-layer Ruston turbine(RT)impeller is used,providing a strong shear force to mix the aqueous and organic phases.The length and width of the mixing chamber are 120 mm,and the height is 140 mm.The impeller installation height is 40 mm,diameter is 60 mm,where the disc diameter is 40 mm,blade length is 20 mm.The height of the lower phase port is 28 mm,which is less than the height of the impeller and baffle 1.The organic phase outlet is higher than the aqueous phase outlet.

Fig.1.Schematics of (a) concurrent mixer-settler and (b) countercurrent mixer-settler.

The entire computational domain was meshed with a hexahedral mesh to enhance the stability of the simulation and to speed up the calculation.The minimum Orthogonality is above 0.56.In addition,due to the large gradient of variable variation in the mixing chamber and the entrance/exit area,the meshes in these areas were densified.The final mesh is shown in Fig.3(b).

3.Numerical Simulation

3.1.Multiphase model

In this study,three different phases are involved,namely the aqueous phase,the organic phase and the gas phase.The Eulerian multiphase flow model was employed to simulate the flow behavior,and each phase is considered as interpenetrating continua.In this view,the same pressure equations are shared between the different phases,but the continuum and momentum equations are solved separately for each phase and then coupled by interaction forces.In addition,the set of equations is closed by an additional volume fraction equation.In countercurrent mixer-settlers,the different phases are uniformly dispersed in the mixing chamber region and then separated in the settling chamber.Given this,the Eulerian model can better describe these behaviors.Its continuum and momentum equations can be expressed as:

wherep，qis the phase index,α，ρ，v denote the volume fraction,density and velocity of phaseq,respectively.Pis the pressure shared by all the phases.τqis the stress strain tensor of phaseqand is defined asgdenotes the acceleration of gravity.Fpqdenotes the interaction force between phasepand phaseq.

The interaction forces mainly consist of drag force,lift force and virtual mass force.The latter two are of very small magnitude in this system,so they are often neglected in the process of calculation [29-31].When only the drag force is considered,the formula can be expressed as:

whereCDis the drag coefficient and the expression varies with different models.In the present study,CDis calculated using the correlation of Schiller-Naumann model [32],as given below

whereReis the relative Reynolds number and is defined as:

wheredpis the diameter of the bubbles or droplets of phasep,μqis the viscosity of the phaseq.

3.2.Turbulence model

In the present work,the Reynolds time averaging principle was applied to the solution of the Navier-Stokes equation.The Reynolds stress term is generated when solved by the Reynolds time-averaged method,and this term cannot be solved directly at present.Therefore,the equation can only be closed by building a turbulence model to relate it to the variables that can be obtained.Thek-ε model[33],k-ω model[34]and Reynolds stress model [35] are commonly used for such a system.Although the Reynolds stress model is based on anisotropy and is suitable for rotational flows,it also has the disadvantages of poor stability,a tendency to diverge in the process of solution,and high computational resource expenditure.After the previous study [36],it is found that thek-ω model does not differ much from thek-ε model in solving,but thek-ω model requires more computational resources.Thus,thek-ε model is chosen as the turbulence model in this paper.The Standardk-ε model is not so effective in solving rotational flows due to its isotropic assumption [37],while the RNGk-ε model [38] takes into account the effect of swirl flow and the expression is

Fig.2.Schematic of the countercurrent mixer-settler.

Fig.3.(a) Geometric dimensions of the countercurrent mixer-settler;(b) perspective view of the mesh.

where the subscript m represents the mixed phase.kis the turbulent kinetic energy.ε is the energy dissipation rate.Gkrepresents the generation of turbulence kinetic energy due to the mean velocity gradients,calculated as:

vi，vjrepresent velocity magnitudes in different directions.μeffrepresents the effective viscosity,calculated as:

The rotation modification was performed in this work and is represented as follows:

where Ω is a characteristic swirl number;αsis a swirl constant.

Cμ，C1ε，，Ω，αsare the model constants and the values are listed in Table S1(see Supplementary Material).

3.3.Fast Fourier transform

The FFT method is employed to analyze the time-varying characteristics of local regions of the flow.Firstly,the data on the variation of the flow in the local region with time are derived from CFD simulations.Then an FFT is performed to obtain the power spectrum,as follows.

For the flow datay（l）over a period of time,the frequency domain datay（θ）is obtained by performing FFT according to the following equation:

whereWn=e（-2πi）/n,nis the number of sampling points;land θ are both sequence indices.

Typically,the frequency data is in the form of complex numbers.In order to express its meaning,it needs to be squared to obtain the power spectrum density (PSD),and its corresponding period is the ratio of sequence number to sampling points,as follows:

3.4.Numerical details

In this study,the whole computational field was divided into three domains:two rotational domains near the impellers and stationary domain in the remaining area.The SMM was used to characterize the rotation of impeller for transient simulations.The geometric model contains two inlets and two outlets,while the top is also set as an outlet because it is connected to the atmosphere.Both inlets were set as velocity inlets,and the three outlets were set as pressure outlets.The inlet velocity of organic phase is 0.07 m·s-1and the inlet velocity of aqueous phase is 0.21 m·s-1.The rotating speed is 300 r·min-1.The detailed boundary conditions can be found in Fig.4.

In countercurrent mixer-settlers,droplets and bubbles are dispersed in the aqueous phase in the mixing chamber and then separated by settling in the settling chamber.Therefore,in this study,the aqueous phase was set as the continuous phase and the organic and gas phases were set as the dispersed phases.In the present work,Ansys Fluent software is employed to solve all mathematic equations.Considering the intense transient variation in the countercurrent mixer-settler,transient solver is applied in this simulation.The SIMPLE Coupled Phases algorithm is used to solve the pressure-velocity coupling format.Spatial gradient is discretized using a least squares based approach.In order to ensure the high accuracy of the calculation,the variables are discretized in QUICK format with third-order precision,except for the pressure,which is discretized in PRESTO!.The time step is set by an adaptive method with the constraint that the global Courant number is less than 1.

4.Results and Discussion

4.1.Validation of the model

To reduce the influence of the number of meshes on the results,four sets of meshes were designed for mesh independence validation,which are very fine (3,839,341 cells),fine (2,846,530 cells),medium (1,561,083 cells),coarse (840,401 cells).Since the mixing chamber simulation is the most critical part,the mesh independence analysis were performed by examining the distribution of velocity in the mixing chamber region at both the cross sectionZ=40 mm as well asZ=120 mm.As shown in Fig.5.It can be found that the simulation results of the four meshes are relatively close,except for the coarse mesh.Considering the computational speed,medium mesh was used for the simulation in this work.

The validity of the gas-liquid-liquid three-phase model and the turbulence model applied in this work also needs to be confirmed before proceeding to the formal simulation.Since this study system contains both mixing and settling chambers,and the flow states in these two chambers are completely different.In the mixing chamber,the flow is mainly violent swirl flow,while in the settling chamber,the flow has a small Reynolds number.The relevant parameters in both chambers need to be verified simultaneously.Since basically few people have studied mixing and settling chambers together,we refer to the literature of experiments related to mixing and settling chambers respectively,to model and mesh their experimental equipment.Then we substitute the multiphase flow model and turbulence model in this study,and finally compare this simulation with their experimental results to verify the validity of the models (the relevant simulations can be found in Fig.S1,in Supplementary Material).For the mixing chamber,Svenssonet al.[39] studied a mixing device with a similar mixing chamber as ours,which uses a Ruston impeller and a kerosene and water system.Since swirl flow is the main feature in the mixing chamber,we compared the ratio vr/vtipof radial velocity to blade tip velocity and va/vtipof axial velocity to blade tip velocity,and the results are shown in Fig.6(a),indicating a good agreement between experimental and simulation results.For the settling chamber,a gravity settling tank was investigated to study the settling and separation characteristics of water and tri-butyl phosphate by Thakeret al.[2,40].Since the degree of water and organic phase separation in the settling chamber is its main evaluation index,we examined the dispersed phase distribution at different heights,and the comparison results are shown in Fig.6(b),indicating that the experimental and simulated results are in good agreement.By comparing the mixing chamber and the settling chamber,it can be determined that the present CFD model is applicable to this study system at the same time.

Fig.4.Details of the boundary conditions.

Fig.5.Velocity distribution for different number of grids at (a) plane Z=120 mm and (b) plane Z=40 mm.

4.2.Global flow pattern analysis

4.2.1.Dispersed phase distribution

The organic phase distribution can directly reflect the degree of liquid phase mixing and separation.Fig.7 shows the organic phase distribution at different moments.At the initial moment,the volume fraction of the organic phase in the mixing chamber is set to 0.5,as shown in Fig.7(a).This setting is close to the situation at reality,which allows the calculation to reach stability quickly.As time proceeds,it can be seen that the distribution of the organic phase in the tank changes markedly.In the mixing chamber,The organic phase gathered at the stirring bar and at the top of the mixing chamber,while the aqueous phase gathered directly below the stirring bar and near the baffle 1.This is due to the fact that in these areas the velocity is low and the shear force is not sufficient causing the water and oil to be difficult to mix,forming the socalled dead zone.By optimizing the structure of the tank and the type of impeller the extent of the dead zone can be reduced and the mixing performance enhanced.At the same time,the dispersion band gradually forms at the intersection of the mixing and settling chambers,and the separation of water and oil mainly occurs in this region.The smaller the thickness of the dispersion band,the faster the separation is indicated.At 15 s,as shown in Fig.7(d),the simulation reached stability,the organic phase distribution in the tank basically ceased to change,and the inlet and outlet flow rates reached equilibrium.

4.2.2.Pressure distribution

Fig.8(a) shows the pressure distribution at theY=0 mm cross section.It can be seen that there is a clear difference in the pressure distribution in the different chambers.The left settling chamber obviously has a wider pressure distribution,while the mixing chamber and the right settling chamber have a narrower pressure distribution.Comparing the organic phase distribution in Fig.7,it can be seen that the left settling chamber is dominated by the organic phase,while the mixing chamber is dominated by the mixed phase.When the pressure at the bottom of the two chambers is the same,the difference in density results in a difference in liquid level,which leads to the pressure distribution shown in Fig.8(a).The right settling chamber is separated from the mixing chamber by a baffle,so the liquid level is close,but the bottom pressure is higher because it is dominated by the aqueous phase.At the same time,the impeller rotation makes the pressure in the central region of the mixing chamber lower compared to the other regions.Fig.8(b)-(e) show the pressure distributions in different height sections of the mixing chamber,which also proves the fact.

Fig.6.The comparison of experimental data and simulation data: (a)vr/vtip,va/vtip at plane Z=10 mm;(b) dispersed volume fraction at plane Y=105,80,60 mm.

Fig.7.Distributions of organic phase at different moments: (a) t=0 s;(b) t=5 s;(c) t=10 s;(d) t=15 s.

Fig.8.Static pressure (mixture) distributions at (a) Y=0 mm;(b) Z1=120 mm;(c) Z2=80 mm;(d) Z3=40 mm;(e) Z4=25 mm.

4.2.3.Macro-instabilities analysis

Nurtono [41] studied the flow patterns in a stirred tank equipped with a double RT impeller,and three typical patterns were summarized.These three patterns alternate with time,which is known as MI.As shown in Fig.9(a),Nurtono’s study found that the variation of the flow pattern mainly lies in the mixing state of the fluid at the bottom of the stirred tank and the flow direction near the stirring blades.

A similar MI was found for the countercurrent mixing and settling tank.Fig.9(b) lists the flow patterns for three types of countercurrent mixing and settling tanks.It can be seen that the velocity vector near the blades changes alternately from parallel to the radial direction to the axial direction of the offset,and this offset affects the position and shape of the surrounding vortex.At the same time,the degree of fluid crossover directly below the impeller also changes continuously.These alternating phenomena make the flow in the mixing tank more complex.

Fig.9.Typical flow patterns of (a) a regular stirred tank [41] and (b) the countercurrent mixer-settler.

4.3.Flow periodicity analysis

Fig.10(a) and (b) show the instantaneous flow variation at the phase port between the mixing and settling chambers,respectively.It is intriguing that the variation shows a trend of periodic fluctuations.In order to verify the periodicity,the FFT method is adopted.Since the initial conditions cannot be set exactly close to the real situation,this leads to a dramatic change in the flow field at the beginning.Therefore,the first 3 s of data were discarded.The FFT was performed only on the later data to obtain the periodogram,as shown in Fig.10(c) and (d).The horizontal coordinate represents the period and the vertical coordinate represents PSD.A higher PSD indicates a better match for that period,but multiple high PSD periods indicate a weaker periodicity.

Fig.10(c)shows the periodogram of upper phase port.It can be seen that there is only one high PSD period of 1.2 s,indicating that the flow of upper phase port fluctuates according to a period of 1.2 s.Fig.10(d) shows the periodogram of lower phase port.It has two high PSD periods of 1 and 1.2 s,both of which are possible.The reason for this phenomenon is:(i)the 1.2 s cycle is due to the liquid in the mixing chamber,which is the same step as above;(ii)the 1 s cycle is due to the liquid in the settling chamber 1,because of the equilibrium formed between the inlet water and the oil outlet.

Similarly FFT was done for the aqueous phase outlet and the organic phase outlet.Fig.10(e) shows only one high PSD for the aqueous phase outlet,again 1.2 s,which is consistent with the period of upper phase port,indicating that the main effect comes from the mixing chamber.However,Fig.10(f) shows more disturbing cycles,which may be due to the change of fluid at the organic phase exit from full organic phase to oil and air.These disturbances would be reduced if better initial conditions were given so that oil and air were present at the outlet at the beginning.

4.4.Phase port flow pattern analysis

The flow pattern at the phase port is further analyzed.Fig.11(a)shows the velocity vector distribution at this location,where two velocity vectors with opposite directions are present.This indicates that the fluid in the mixing chamber is flowing into the clarification chamber at the same time that the fluid in the clarification chamber is flowing into the mixing chamber,and that the boundary between these two fluids is very clear.The reason for this phenomenon can be attributed to the difference in fluid flow velocities between the clarification and mixing chambers,and this velocity difference generates vortices at the phase port,which causes the two fluids to be distributed on both sides of this interface.This distribution is related to the direction of rotation of the impeller.

TheX-direction velocity vector at this position can more clearly characterize this distribution.If the vector along theX-positive direction is marked as ‘‘+” and the vector in theX-negative direction is marked as‘‘-”.Then as shown in Fig.11(b)and(c),the vectors at these two phase ports will vary in distribution with time.When the ‘‘+” flow is more,the macroscopic flow presentsM＞0,and when the ‘‘-” flow is more,the macroscopic flow presentsM＜0.If we compare this phenomenon with the periodicity discussed in the last section,we will find that the variation is consistent with it.

Fig.10.Instantaneous flow rates of(a)upper phase port and(b)lower phase port.Periodograms of(c)upper phase port;(d)lower phase port;(e)aqueous phase outlet;(f)organic phase outlet.

The reason for this periodicity can be further attributed to the fact that the vortex has an energy redistribution function.The motor transfers kinetic energy to the fluid through the impeller.As this fluid passes through the phase port,the vortex redistributes its kinetic energy,which causes some of the kinetic energy to enter the settling chamber and some to return to the mixing chamber.When the kinetic energy accumulates to a certain level in the mixing chamber,it will be released,and this process will continue to cycle,thus creating periodicity.

Since the size and intensity of the vortex is related to the composition of the liquid in the mixing chamber and the speed of the impeller,the period at the phase port can reflect the composition of the liquid phase in the mixing chamber when the speed of the impeller is constant.The monitoring of the liquid phase composition can be achieved by establishing the relationship between the period length and the ratio of the liquid phase composition.The analysis in the previous section shows that the period at the outlet of the tank is the same as the period at the phase port,so the liquid phase composition in the mixing chamber can be determined by FFT analysis by detecting the flow rate change at the outlet.

Fig.11.(a) Velocity vector distribution of the phase ports;distributions of X-velocity at (b) the lower phase port and (c) the upper phase port under different flow rate.

5.Conclusions

In this study,the mixing and separation processes of fluids in countercurrent mixer-settlers were investigated by CFD methods.A model with universality was developed.Based on this model,the following conclusions are drawn:

(1)Local dead zones are formed in parts of the mixing chamber due to the characteristics of the double Ruston impeller.Besides,dispersion bands are formed in the settling chamber,whose width determines the separation speed.Future optimization can be put into changing the impeller type as well as the baffle position.

(2) The macroscopic instability analysis of the flow pattern shows that the presences of the settling chamber have little influence on the flow pattern of the mixing chamber.

(3) The period of outlet flow can be determined by fast Fourier transform method.Therefore,by monitoring the fluctuation of outlet flow and then analyzing with FFT method,it is possible to determine whether the whole tank is in normal condition and quickly get the cause of abnormal condition.This method is very promising in control systems.

(4) Two types of velocity vectors in opposite directions are distributed on the left and right sides of the phase port,which indicates that the material exchange in the mixing and settling chambers is simultaneous,thus explaining why the level in the mixing chamber can remain stable.

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

In this work,we sincerely appreciated the National Natural Science Foundation of China (21978198) for financial support.

Supplementary Material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cjche.2021.10.001.

Nomenclature

Aiinterfacial area,m-1

CDdrag coefficient

Cμ，C1ε，model constants

dpdiameter of the bubble or droplet,m

Finteraction force,N

fdrag function

Gthe generation resume of turbulent kinetic energy

ggravitational acceleration,m·s-2

lsequence index

Mmass flow rate,kg·s-1

Nimpeller agitation speed,r·min-1

Pessure,Pa

ReReynolds number

v velocity,m·s-1

vi，vjvelocity magnitudes in different axes,m·s-1

vrradial velocity,m·s-1

vaaxial velocity,m·s-1

vtiptip velocity,m·s-1

tflow time,s

WnFFT calculation formula

Yfrequency domain data

y（l） flow data

α volume fraction

αsswirl constant

ρ density,kg·m-3

μ viscosity,kg·m-1·s-1

μeffefficient viscosity,kg·m-1·

ε energy dissipation rate,m2·s-3

γ surface tension coefficient,N·m-1

θ sequence index

τ stress strain tensor

ζ particulate relaxation time

ω specific dissipation rate,s-1

Ω characteristic swirl number

Subscripts

mmixer phase

p，qphase index