Zhijian Zuo,Shuguang Gong,Guilan Xie

School of Mechanical Engineering,Xiangtan University,Xiangtan 411105,China

Keywords:Intensive mixer Particle mixing DEM Mixing performance Power consumption

ABSTRACT Discrete element method (DEM) simulations of particle mixing process in an intensive mixer were conducted to study the influence of structural and process parameters on the mixing performance and power consumption.The DEM model was verified by comparing the impeller torque obtained from simulation with that from experiment.Impeller and vessel torque,coordination number (CN) and mixing index(Relative standard deviation) were adopted to qualify the particle dynamics and mixing performance with different parameters.A method based on cubic polynomial fitting was proposed to determine the critical mixing time and critical specific input work during the mixing process.It is found that the mixing performance and energy efficiency increases with the decrease of impeller offset.The mixing performance is improved slightly with the increase of blade number and the impeller with 3 blades has the highest energy efficiency due to its low input torque.Results indicate that the energy efficiency and the mixing performance increase with the decrease of filling level when the height of granular bed is higher than that of blade.

1.Introduction

Granular mixing is an important operating unit in industries such as nuclear,pharmaceutical,coal,and mining,et al.[1].Unlike fluid flow with governing equations,granular flow has no constitutive,it depends on several parameters and each particle moves distinctly,thus,the method of improving mixing performance of mixer attracted more and more researchers’ interest [2].Intensive mixer is widely used in granular mixer due to its high efficiency and simple construction.Proper structural and process parameters are very important to improve the mixing performance of mixer.However,it is very difficult to select proper parameters and predict the mixing time accurately due to the mixing mechanism is imperfectly understood up to now.In fact,long mixing time is usually adopted in engineering to obtain an ideal mixing effect,which results in a huge waste of energy.On the other hand,in the process of mixer design,engineers mainly depend on the experiment and experience due to lack of scientific knowledge,leading to the designed mixer fails to meet the requirement of optimal design and energy saving.Obviously,the description of granular flow and the optimal design of mixer are two challengeable works in the granular mixing industry [3].

Most knowledge about the mixing process of granular matter is obtained from experiments and numerical simulation.Over the years many researchers have invented various experimental methods and devices to investigate the particle movement and mixing performance.Nadeem presented an overview of the non-invasive method for describing granular mixing in recent decades [4].The most commonly used methods in mixing process monitoring are particle image velocimetry (PIV) [5,6],positron emission particle tracking (PEPT) [7],and torque measurement [8].Experimental research has significantly improved the development of mixer design,but the traditional intrusive measurement methods will interfere with the system due to the characteristic of granular matter such as compactness,changeable structure and strong interaction et al.and the new non-intrusive methods cannot meet the needs of dynamic measurement as the penetration capacity and temporal and spatial resolution.Besides,experimental research also needs to consider the problems of long period and high cost.

In recent years,with the development of computer technology,numerical simulation has become an important method to study granular flow due to its strong adaptability,repeatability,and low cost.As a robust and precise numerical method,the discrete element method (DEM) solves the force and motion of particles on a single particle scale,so it has unique advantages in studying the particle dynamics and mixing mechanism of granular matter[9].Many researchers have analyzed the mixing performance and particle dynamics in rotary drum [10,11],bladed mixer [12,13],spouted bed[14,15]and static mixer[16]based on DEM,Zhu presented a comprehensive overview about the DEM application in industrial fields [17].Gong et al.analyzed the influence of particle properties and the impeller speed on the mixing performance of intensive mixer [8].Bao et al.analyzed the effects of blade configuration and process parameters on granular mixing in a cylindrical mixer using DEM[18].Boonkanokwong studied the effect of granular properties and operating parameters on the impeller torque and power consumption of a bladed mixer[19].On the other hand,dynamics of non-spherical particles during the mixing process were analyzed and results show that the shape of particles has an important influence on the mixing performance of granular matter [20,21].The influence of liquid fraction on the mixing process of particles in the rotary drum [22,23] and bladed mixer[24,25]was investigated based on DEM and obtained results show that a small amount of liquid between particles will decrease the mixing performance.DEM can provide abundant micro-scale information and it plays an important role in understanding the structure of granular material,internal physical processes,and insights into new phenomena.The intensive mixer has been widely used in industries,but there is little literature focused on the study of the influence of structure and process parameters on the mixing performance and energy consumption.

In this work,DEM simulations based on EDEM software were conducted to study the influence of structural and process parameters on the influence of mixing performance and power consumption of intensive mixer.The simulation model was verified by comparing the impeller torque of simulation with that of the experiment.Then we analyzed the influence of impeller offset,filling level,and blade number on the mixing performance and power consumption of intensive mixer.Specially,we focused on the input torque,coordination number (CN),and the relationship between mixing index and mixing time,specific input work during mixing process with different parameters.A method based on cubic polynomial fitting was proposed to determine the critical mixing time and critical specific input work,which are very important parameters in the large-scale industrial mixer.Results obtained in this work will be helpful for the improvement of the intensive mixer design to high efficiency and low energy consumption.

2.Numerical Setup

2.1.Discrete element method

The DEM proposed by Cundall solves the force and motion of a single particle based on Newton’s equations.The governing equations of a particle i in this method can be described as

where,mi,vi,Ii,Riand ωiare the mass,velocity,moment of inertia,radius,and angular velocity of particle i.g denotes the gravitational acceleration.

A proper contact model has a significant influence on the accuracy of simulation results.In this study,Hertz-Mindlin with no-slip contact model was adopted due to its accuracy and efficient force calculation in dry particle simulation.In this model,the normal force component is based on Hertz’s theory,and the tangential force is based on Mindlin’s work.Both normal and tangential forces have damping components [26].The normal and tangential forces between particle j to i are given by

where knand ktare the stiffness coefficient in the normal and tangential direction.cnand ctare the damping coefficient in the normal and tangential direction.δij,nand δij,tare the overlap of normal and tangential direction between particles i and j.these parameters are calculated as follows:

where E*,R*and m*are the equivalent Young’s Modulus,equivalent radius,and equivalent mass,respectively.Snand Stare the normal and tangential stiffness,respectively.

2.2.DEM parameters

The choice of DEM parameters has great influences on the results of simulation.Material property and contact model are two kinds of parameters in DEM simulation with Hertz-Mindlin no-slip model.

In this work,iron concentrate particles were used to study the mixing performance and power consumption of intensive mixer.The density of particle is 5200 kg﹒m-3,which is obtained from iron mine.The particle size distribution was measured by vibrating screen,and the result is shown as Fig.1.Considering the maximum diameter measured is about 9.5 mm,the average particle diameter can be estimated as below:

Fig.1.Particle size distribution of particles.

the obtained average diameter is 3.46 mm.If the particle diameter is set to 3 mm,it is too time consuming to simulate the approximately 0.4 million particles with current computational capability.Remy et al.[27] found that the particle size in cohesionless flows has little effect on the principal flow pattern in a bladed mixer.So,we set an particle diameter with 4 mm in simulations.Carbon steel Q235B was used as the material of mixer.On the other hand,the contact parameters include coefficient of restitution,static,and rolling friction.In this work,the stacking angle experiment using granular property tester (BT-1000,BETTERSIZE) shown as Fig.2 was conducted to measure the repose angle φ,the coefficient of static friction can be obtained as

the obtained repose angle for particle–particle and particle–wall are 27° and 24°,the corresponding static friction coefficients are 0.5 and 0.45,respectively.

The rolling friction is set to be a small value due to it is very hard to measure and it plays a weak influence on the simulation[28].In this work,it was set as 0.02 for both particle–particle and particle–wall.The restitution coefficient is obtained by a high rebound test and the measured data are 0.65 and 0.6 for particle–particle and particle–wall,respectively.Detailed parameters of material property and contact model are presented in Table 1.

2.3.Geometric parameters

A schematic of the intensive mixer used in this work is described in Fig.3,which has the same size as the experimental apparatus used in METALLURGICAL CORPORATION OF CHINA LTD.(MCC).It can be obtained that the intensive mixer is characterized by an offset between vessel and impeller,and the impeller and vessel rotate inversely at the same time.The dimensions and parameters of the intensive mixer are listed in Table 2.

3.Results and Discussions

3.1.Modeling verification and mixing process

The experiment apparatus used in this work is an intensive mixer test rig shown as Fig.4.The rig is made up of vessel,impeller,body,and control cabinet.Vessel and impeller are driven by two motors respectively.A torque sensor (FORSENTEK FY01)between the impeller and vessel is used to measure the impeller torque during the mixing process.The equipment has the same size as the simulation model detailed in Table 2.After vibration screening,particles with a diameter ranging from 3 to 5 mm were used for the experiment.Particles with similar size and properties detailed in Table 1 were adopted in DEM simulation.DEM model was validated by comparing the impeller torque obtained fromsimulation and that from experiment shown as Fig.5.The dotted line in the figure denotes the average torque during the mixing process.It can be obtained that the impeller torque in simulation and experiment fluctuate during the mixing process,and the average impeller torque for experiment and DEM simulation is 1.74 and 1.71 N﹒m,respectively,which indicates the results of simulation and experiment are consistent with each other.The average torque of experiment is slightly larger than that of simulation due to the shape of particles used in the experiment is irregular,while the shape of particles used in the simulation is regular sphere,the flowability of particles with regular spherical shape is better than particles with irregular shape,leading a larger torque during the mixing process.A small fluctuation amplitude ranging from 1.42 to 2.1N﹒m can be seen in the experiment,while the torque fluctuation amplitude ranging from 1.05 to 2.98 N﹒m in simulation is larger than that in experiment.The fluctuation frequency of simulation torque is also larger than that of experiment torque.This is because the impeller speed in simulation is a constant,while the impeller speed in experiment changes with impeller torque due to the motor used to drive impeller is an ordinary but not a high precision inverter motor.

Table 1 Simulation parameters

Table 2 Dimensions of intensive mixer

The schematic of the mixing process in an intensive mixer at the top view is shown as Fig.6.The speed of impeller and vessel are set as 100 and 10 r﹒min-1.The filling level of mixer is 60%,and there are about 74,460 particles placed in the vessel.An equal number of red and black particles are loaded side by side in the vessel at the initial stage.The phenomenon of particles mixing in impeller area and particle flow under the rotation of vessel can be obtained by studying these pictures.Particles near the impeller move violently and exchange positions with each other under the impact of high-speed impeller,particles exhibit tangential movement related to the motion of impeller.the mixing degree increases mainly depends on the mixing process in the impeller area.On the other hand,under the low-speed vessel,particles outside the impeller region are constantly mixed with shearing and diffusive motion,the well-mixed particles are moved out of the impeller area,and poorly mixed particles are sent into the impeller area for mixing.The distribution of particles becomes more and more uniform with this circulation process repeatedly.

Fig.2.Repose angle measurement:(a) test equipment;(b):experiment.

Fig.3.Schematic of intensive mixer.

Fig.4.Mixing experiment.(a) equipment;(b) control cabinet.

3.2.Influence of impeller offset

Fig.5.Impeller torque of simulation (a) and experiment (b).

During the mixing process in an intensive mixer,particles mixed rapidly under the impact of high-speed impeller,lowspeed vessel sends poorly mixed particles into the impeller region and move mixed particles out of the impeller region[8].The offset between vessel and impeller has an important influence on the mixing performance and particle dynamics due to the impact of impeller on particles.During the mixing process of intensive mixer,the impeller has an influence area shown as Fig.7,in which particles move violently under the action of high-speed rotating impeller.The diameter of influence area (D1) is dependent on the impeller diameter and speed,and it can be estimated from the mixing process snapshot in Fig.6.In order to avoid the dead zone shown as Fig.8,the impeller offset and the vessel diameter should meet the following relationship:

In this work,simulations with offsets of 35,55,and 75 mm shown as Fig.9 were performed to qualify the influence of offset on the mixing performance of intensive mixer.In order to avoid the dead zone,the impeller influence area should cover the entire vessel under the rotation of vessel,and the corresponding blade diameter was set as 200,160,and 120 mm,respectively.The speed of impeller and vessel was set as 100 and 10 r﹒min-1in the simulations.

Fig.6.Schematic of the mixing process in an intensive mixer.

Fig.7.Schematic of impeller influence area.

The torque required in granular mixing depends on several factors,including filling level,impeller structure,rotational speed,and mixer size.Since torque is sensitive to changes in particle mixing,it can be used as a parameter to probe the efficiency of mixer design.The intensive mixer has two power inputs:impeller and vessel.Fig.10 shows the evolution of input torque for impeller(top) and vessel (bottom) with different impeller offset.It can be observed that both impeller torque and vessel torque fluctuates during the dynamic equilibrium stage.The average impeller torque for the offset of 35,55,and 75 mm is 2.83,1.71,and 0.98N﹒m,respectively,which means the impeller torque increases with the decrease of impeller offset.When getting mixed,particles exert a load on blades and give rise to the agitation torque.Small offset has a large diameter impeller,which has more contacts with particles in the mixing process,leading to a large impeller torque.It also can be obtained that the average vessel torque for the offset of 35,55,and 75 mm are 3.04,2.06,and 1.62N﹒m,respectively,indicating the vessel torque also increases with the decrease of offset.In fact,a large impeller intensifies the movement of particles along the blade direction,and the direction of vessel movement is opposite to that of impeller,which leads to the increase of vessel torque during the mixing process.

The coordination number (CN) is an important parameter for particle flow due to it reflects the compactness of particle bed.For a single particle,its CN is defined as the total number of other particles in contact with it.In a randomly packed particle system,the coordination number of each particle is different and the average coordination number of all particles is usually used as the coordination number of particle systems.A large coordination number means small space during particles and particles move hardly in the mixing process.CN can be described as [29]

where Ncand Npare the total contact number and particle number in the mixer,respectively.

Fig.8.Schematic of dead zone:(a):changing diameter separately;(b):changing offset separately.

Fig.9.Schematic of intensive mixer with different impeller offset.

Fig.10.Evolution of torque input for different impeller offset:(a) Impeller.(b)Vessel.

Fig.11 presents the evolution of CN with different impeller offset.It can be observed that the CN has a peak in the initial stage due to a static package state.Then it decreases rapidly under the impact of high-speed impeller.After that,steady particle flow formed slowly under the regular rotation of impeller and vessel,the coordination number increases gradually at the same time and finally reach dynamic equilibrium stage.The average CN for the offset of 35,55,and 75 mm in dynamic equilibrium stage are 2.78,3.81,and 4.5,respectively,which means increasing the offset(with decreasing the blade diameter) increases the space between particles mixing.This is due to small offset with large diameter impeller intensifies the movement of particles,leading to a small CN during the mixing process.

Fig.11.Evolution of CN for different impeller offset.

Mixing performance is an important indicator during the mixing process.So far,mixing degree has been used to assess the mixing performance and lots of mixing indexes were proposed by various authors such as segregation index,Lacey index et al.The mixing time is usually used to characterize the mixing performance.Relative standard deviation(RSD),one of the most common mixing index,was adopted to qualify the mixing performance and in turn the efficiency of mixing process in an intensive mixer.The proportion of particle A in a computational cell I is

where NA,Iand NT,Iare the number of particle A and total particle number in a computational cell I,respectively.

The average proportion of particle A in the total mixture is

Here n is the number of computational cells.

Then the Relative standard deviation for particle A can be given as

The smaller the S,the better the particles mixed.

The calculation of mixing index requires sampling and the sampling size has an important influence on the computational results.Chazal and Hung [30] found that the results may fail to reveal the full extent of mixing when samples containing too few particles,and the mixing index could be independent of sample size if samples contain particles more than some ten thousand.Samples should be taken at different points to give a global view of the mixing process,and they cannot overlap to avoid multiple counting of particles.Arntz [31] recommended that good results can be obtained if there are 50–70 particles in each sampling cell.In this work,cubic sampling grids were adopted to divide the intensive mixer shown as Fig.12.There are 1944 (18 × 18 × 6) cube cells arranged in the computational area,and approximately 64 particles are placed in each sampling cell.

Fig.13 shows the influence of impeller offset on the relationship between the mixing index and mixing time.The initial value of the mixing index for different offset is 0.61 due to three simulations have the same initial packing stage.Then it decreases with the regulation movement of impeller and vessel.Considering the case of the offset with 55 mm,the mixing process can be divided into three stages:rapid mixing stage,slow mixing stage,and dynamic equilibrium stage.In the rapid mixing stage,the mixing index decreases rapidly from 0.61 to 0.463 at 10 s.Then it decreases relatively slowly from 0.463 to 0.428 in the 20 s in the slow mixing stage.After that,the mixing index entry a dynamic equilibrium and it fluctuates around 0.415.These trends are similar to the results presented by German [32].However,the mixers with different impeller offset have different mixing performance.The mixing index for the offset of 35,55,and 75 mm at 10 s are 0.43,0.465,and 0.525,respectively,and it declined to 0.41,0.428,and 0.468 at the 20 s,indicating mixing performance increases with the decrease of impeller offset.As mentioned above,a small impeller offset mixer with a large diameter blade enhances the movement of particles,leading to better mixing performance in the mixing process.A comparison of the mixing index in Fig.13 and CN in Fig.11 show that particle compactness influences its mixing process.

The mixing time is an important but not unique characteristic of mixing performance.In many situations,the input power is also a key indicator to measure the mixing performance.In this work,Specific input work [33] was used to qualify the power consumption during the mixing process.The equation of specific input work can be described as:

where Winputdenotes the input power per unit volume.Tiand Tvare the torque of impeller and vessel,respectively.niand nvare the rotational speed of impeller and vessel.V is the total particle volume in the vessel.

Fig.12.Schematic of the sampling grid.

Fig.13.Influence of impeller offset on the relationship between mixing index and time.

Fig.14 presents the influence of impeller offset on the relationship between the mixing index and specific input work.Considering the overall mixing process,the mixing index decreases with the increase of specific input work,which means particles are mixed continuously with external input work.It can be observed that the mixing index is about 0.61 at the initial stage for three simulations with different impeller offset.After that,the mixing index with an offset of 35,55 and 75 mm declined to 0.49,0.52 and 0.53 respectively when the specific input work is 50 kJ﹒m-3,and then the mixing index declined to 0.448,0.458 and 0.465 with a specific input work of 100 kJ﹒m-3.At the same specific input work,a lower relative standard deviation means better energy efficiency.The results show that the energy efficiency of intensive mixer increases with the decrease of impeller offset.This is due to the high-speed movement of particles in the mixer with small offset and long blade,convective mixing is dominant in the mixing process,leading to higher mixing efficiency.

Fig.14.Influence of impeller offset on the relationship between the mixing index and specific input work.

Fig.15.Schematic of critical mixing time.

According to the analysis above,the mixing degree of particles does not increase with the increase of mixing time after reaching the dynamic equilibrium stage,which means that it will waste energy if the mixing time is too long.On the other hand,the mixing performance cannot meet the requirements when the mixing time is too short.The critical mixing time and critical specific input work are two very important parameters during the large-scale industrial mixing process.So far,many methods have been proposed to determine the critical mixing time under different conditions [34,35].In this work,the obtained relationship between mixing index and mixing time is investigated by curve fitting.Comparing various forms of curve fitting,it is found that the cubic polynomial fitting has the smallest error and highest reliability.The critical mixing time tcis determined by .s0=λ0shown as Fig.15,where s is the cubic polynomial fitting of the mixing index,λ0is the slope of curve on the critical mixing time or critical specific input work and it is commonly set to zero or a very small value.In this work,λ0is set to be 0.003 and 0.0005,for the calculation of critical mixing time and critical specific input work.

Fig.16 presents the critical mixing time and critical specific input work for different impeller offset.It can be observed that the critical mixing time for the offset of 35,55,and 75 mm are 13.7,17,and 34.3 s,respectively,which means the mixing performance is increased with the decrease of impeller offset.This is due to particles move more violently under the impact of a blade with small offset and large diameter.On the other hand,the critical specific input work for the offset of 35,55,and 75 mm are 133.9,133.2,and 134.7 kJ﹒m-3,indicating that the impeller offset has little influence on the critical specific input work during the mixing process.

3.3.Influence of blade number

The mixing of particles in the intensive mixer mainly depends on the high-speed rotation of the impeller,which equipped with many blades.The number of impeller blades has an important impact on the velocity and movement of particles in mixer [36].The impeller of the intensive mixer in MCC is composed of three layers of blades shown as Fig.3,and each layer has two blades.In this work,simulations with 1,2,3 blades in each layer shown as Fig.17 were conducted to study the influence of impeller configuration on the mixing performance of the intensive mixer.The corresponding blade number is 3,6,and 9.The offset of impeller in simulations is 55 mm.The speed of impeller and vessel are 100 and 10 rpm,respectively,and the filling level in simulation is 60%.

Fig.16.Critical mixing time and critical specific input work for different impeller offset.

Fig.18 presents the evolution of impeller torque (a) and vessel torque (b) with different number of blades on the impeller.The dotted line is average torque in the mixing process.It can be observed that the average impeller torque for the impeller with 3-blades,6-blades,and 9-blades is 1.09,1.71,and 1.75 Nm,respectively.For a bladed mixer,it was found that higher radial and vertical velocities of particles were obtained in the mixer with one or two blades,which led to a pronounced recirculation flows.The mixer with three and four blades had a larger tangential velocity component of particles.In the intensive mixer [36],which is similar to the bladed mixer,we observed that the impeller torque increases with the increase of blades,this is due to the number of contact between particles and blades increases with the increase of blades.However,the torque for 3-blades is much smaller than that of 6-blades,while the impellers with 6-blades and 9-blades have similar torque.It also can be observed that the amplitude of torque fluctuation increases with the number of blades due to the collision between particles and blades is more violent with the increase of blade.Similarly,the average vessel torque for 3-blades,6-blades,and 9-blades is 1.35,2.06 and 2.13 N﹒m due to it rotates in the opposite direction of the impeller.

Fig.19 presents the evolution of CN with different blade number.It can be observed that three simulations with 3,6,and 9 blades provide similar results regarding the overall curve.However,at the equilibrium stage,the center of fluctuation for 3,6,and 9 blades are 4.02,3.82,and 3.55,respectively,which indicates the coordination number decreases with the increase of blade.This is due to the impact of blades on particle movement increases with the increase of blade number,leading to a small CN for large blade number mixer during the mixing process.

Different impeller has different mixing performance in an intensive mixer.Fig.20 presents the influence of impeller on the relationship between the mixing index and mixing time.it can be observed that the mixing index with different blade number is 0.61 in the initial stage,it declined to 0.488,0.465 and 0.449 at 10 s for the impeller with 3,6 and 9 blades,then it drops to 0.432,0.428 and 0.421at 20 s,respectively.In previous work[36],it was found that the bladed mixer using two or three blades provides better mixing performance than using one or four blades,as evaluated by calculation of the Lacey index of the particle system.Mixer with two or three blades can get a higher granular temperature and particle diffusivities than that with one or four blades.In our work,we observed that the mixing performance increases with the increase of blade number.We hypothesize this is due to the mixing of particles in a mixer is mainly depends on the impact of blades on particles,the impact increases with the increase of blade number,leading to better mixing performance for large number blade mixer in the mixing process.

Fig.17.Schematic of three different impellers.

Fig.18.Evolution of torque input for different blade number:(a) Impeller.(b)Vessel.

Fig.21 shows the influence of blade number on the relationship between the mixing index and specific input work.The results show that for impeller with 3,6 and 9 blades,the mixing index declined from 0.61 at initial stage to 0.502,0.518 and 0.498 respectively when the specific input work is 50 kJ﹒m-3,then it declined to 0.432,0.458 and 0.441 with specific input work of 100 kJ﹒m-3,which indicates the blade number has little influence on the energy efficiency during the mixing process.In fact,as mentioned above,the mixing performance increases with the increase of blade number,however,the impeller torque and vessel torque also increase with the increase of blade number.

Fig.19.Evolution of CN with different blade number.

Fig.20.Influence of blade number on the relationship between mixing index and time.

The critical mixing time and critical specific input work for different impeller blades are shown as Fig.22.The critical mixing time for the blade number of 3,6,and 9 are 18.9 17.1 and 15.5 s,which means the critical mixing time decreases with the increase of blade number.This is because particles move violently in an intensive mixer with more blades.On the other hands,the critical specific input work for impeller with 3,6 and 9 blades are 109.5,133.2 and 117.2 kJ﹒m-3,respectively,which means the impeller with three blades has the highest energy efficiency while the impeller with 6 blades has the lowest energy efficiency during the mixing process.In fact,the impeller and vessel torque of the mixer with 3 blades are less than that of 6 and 9 blades.While the mixing performance increases slightly with the increase of blade number,resulting in the highest energy efficiency for the impeller with 3 blades.

Fig.21.Influence of blade number on the relationship between mixing index and specific input work.

Fig.22.Critical mixing time and critical specific input work for different blade number.

3.4.Influence of fill level

The filling level of particles in a mixer has important effects on the mixing performance [37].Simulations were conducted with 50%,60%,and 70%filling levels to investigate the influences of filling level on particle flow and mixing performance in an intensive mixer,and the corresponding number of particles is 61570,74462,and 87423,respectively.The speed of impeller and vessel is 100 r﹒min-1and 10 r﹒min-1respectively.The offset of impeller is 55 mm.

Analysis of impeller and vessel torque with different filling level is presented in Fig.23.It can be obtained that the average impeller torque for the filling level of 50%,60%,and 70% are 1.32,1.71,and 2.19 N﹒m,respectively.which means the impeller torque increases with the increase of filling level.This is similar to trends that have been obtained by Remy[38]in a bladed mixer.The pressure within the particle bed varies linearly with bed height and can be approximated by hydrostatics.Similarly,the vessel torque for the filling level of 50%,60%,and 70% are 1.59,2.06 and 2.6 N﹒m due to its rotation direction is opposite to that of impeller.Compared to the torque of different filling level,the amplitude of fluctuation with a filling level of 70% is larger than that of 50%.It is because particles continuously move over the impeller blade and fall down in the mixing process,the random impact on the impeller blade increases with the increase of filling level,leading to large torque fluctuation in the mixing process.

Fig.23.Evolution of torque input for different filling level:(a) Impeller.(b) Vessel.

Fig.24.Evolution of CN for different filling level.

Fig.24 shows the evolution of CN with different filling level.It can be observed that simulations with different filling level provide similar results regarding the overall profile of the coordination number.However,the center of fluctuation at the equilibrium stage for the filling level of 50%,60%,and 70% are 3.75,3.85 and 4.01 respectively,which indicates CN increases with the increase of filling level,this phenomenon also can be obtained from the rising stage before the dynamic equilibrium.This is due to the interparticle force around impeller increases with the increase of filler level,leading to low flowability of particles and a small CN during the mixing process.

Different particle dynamics caused by different filling level will affect the mixing performance in an intensive mixer.Fig.25 shows the influence of filling level on the relationship between mixing index and mixing time.It can be observed that simulations with different filling level provide similar results considering the overall curve.the mixing index for filling level of 50%,60%,and 70%decreases from 0.61 in the initial stage to 0.495,0.516 and 0.532 at 10 s,then it drops to 0.432,0.458 and 0.475 at the 20 s respectively,which indicates the mixing index increases with the increase of filling level,and the mixing performance decreases with the increase of filling level.In the previous work of Remy[38],it was found that a three-dimensional recirculation zone develops in front of the blade,which promotes vertical and radial mixing in the bladed mixer.Increasing filling level reduces the size of recirculation zone,decreases bed dilation,and binders particle diffusivities.However,above a critical filling level,the behavior of particles within the span of blade was found to be invariant of filling level.In our work,we observed the mixing performance of particles in an intensive mixer decreases with the increase of filling level.This is due to the recirculation zone and bed dilation decrease with the filling level.

Fig.26 presents the influence of filling level on the relationship between the mixing index and specific input work.The initial mixing index without any input work for different filling level is 0.61,and then it decreases with the increase of specific input work.The mixing index with filling level of 50%,60%,and 70% are 0.495,0.516 and 0.532 respectively when the specific input work increases to 50 kJ﹒m-3,then it declined to 0.432,0.458 and 0.475 with a specific input work of 100 kJ﹒m-3,which indicates the energy efficiency of intensive mixer increases with the decrease of filling level.This is due to particles in the intensive mixer with low filling level have good fluidity,and particles move violently under the movement of impeller and vessel,leading to high energy efficiency in the mixing process.

Fig.25.Influence of filling level on the relationship between mixing index and time.

Fig.26.Influence of filling level on the relationship between mixing index and specific input work.

Fig.27.Critical mixing time and critical specific input work for different filling level.

Fig.27 presents the critical time and critical specific input work for different filling level.It can be obtained that the critical mixing time for the filling level of 50%,60%,and 70% are 14.9,17.1,and 18.7 s,respectively,which indicates the critical mixing time increases with the increase of filling level.This is because the CN of particle bed in a low filling level mixer is smaller than that of high filling level,particles move violently under the movement of impeller and vessel,leading to good mixing performance and little mixing time.It also can be observed the results of critical specific input work are similar with the results of critical mixing time,the critical specific input work for filling level of 50%,60%,and 70% are 108.1,133.2 and 147.6 kJ﹒m-3,respectively,which means the energy efficiency decreases with the increase of filling level.According to the analysis above,the mixing performance increases with the decrease of filling level,while the torque for both impeller and vessel increases dramatically with the increase of filling level,leading to low energy efficiency with high filling level.

4.Conclusions

In this work,DEM simulations were conducted to study the influence of geometric and process parameters on the particle dynamics and mixing performance in an intensive mixer.DEM model was validated by comparing the impeller torque obtained from simulation and experiment.Coordination number,input torque,and Relative standard deviation were used to qualify the mixing performance and particle dynamics during the mixing process.A method based on cubic polynomial fitting was proposed to determine the critical mixing time and critical specific input work.Following conclusions can be obtained according to the simulations:

· The mixing experiment based on an intensive mixer was carried out,and the DEM model was validated by comparing the impeller torque of experiment with that of simulation.

· The mixing performance and energy efficiency increase with the decrease of impeller offset due to the impeller and vessel torque increase with the decrease of impeller offset.

· The mixing performance increases slightly with the increase of blade number.The impeller with 3 blades has the highest energy efficiency due to it has the smallest torque during the mixing process.

· The energy efficiency and mixing performance increase with the decrease of filling level when the particle bed height is higher than that of blade.

The results of our work provide insight into the influence factors during granular mixing in an intensive mixer.In future work,the statistic method will be used to analyze the sensitivity of each factor to the influence of granular mixing in an intensive mixer.Then,the optimization model of granular mixing in an intensive mixer will be established and solved.

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 research is funded by the National Natural Science Foundation of China[51475403],and the financial support to the author is gratefully acknowledged.