Discrete element method study of effects of the impeller configuration and operating conditions on particle mixing in a cylindrical mixer

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Discrete element method study of effects of the impeller configuration and operating conditions on particle mixing in a cylindrical mixer Yuyun Bao a , Tianchi Li a , Dengfei Wang b , Ziqi Cai a,∗ , Zhengming Gao a,∗ a

State Key Laboratory of Chemical Resource Engineering, School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Daqing Petrochemical Research Center, Petrochemical Research Institute, China National Petroleum Corporation, Daqing 163714, Heilongjiang Province, China b

a r t i c l e

i n f o

Article history: Received 29 October 2018 Received in revised form 16 December 2018 Accepted 19 February 2019 Available online xxx Keywords: Cylindrical mixer Impeller configuration Discrete element method Particle Mixing behavior

a b s t r a c t We employed the discrete element method to study the effects of the impeller configuration (i.e., blade diameter, inclination angle, and blade number), rotational speed, and fill level on the flow and mixing of particles in a cylindrical mixer equipped with flat and inclined blades. The coefficient of rolling friction, coefficient of static friction, and coefficient of restitution were experimentally determined before the simulation, and simulation results of the torque and surface particle distribution were validated in experiments, particularly when using a true Young’s modulus in the discrete element method. The performance of the mixer was assessed using the Lacey mixing index. The input work per unit volume was used to represent the mixing efficiency. The circumferential velocity and axial diffusion coefficient of the particles were quantitatively analyzed to reveal the effect of particle flow on the mixing. It was found that the mixing performance and efficiency of a three-blade mixer are better than those of twoand four-blade mixers. For pitched blades, a three-flat-blade mixer has better mixing performance than a three-45◦ -blade own-pumping or a three-45◦ -blade up-pumping mixer, but the mixing efficiency of the three-45◦ -blade up-pumping mixer is the best among these three mixers. As the rotational speed increases, the mixing performance improves but the mixing efficiency hardly changes. When the fill level is 0.4 times the cylinder diameter, the 160D two-flat-blade mixer has good mixing performance with high mixing efficiency. The circumferential velocity has the greatest effect on mixing performance for side-by-side initial loading. © 2019 Published by Elsevier B.V. on behalf of Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences.

Introduction Granular materials are widely used in industrial production and daily life, such as in pharmaceutical processes (Cavinato et al., 2010), chemical engineering (Remy, Canty, Khinast, & Glasser, 2010), metallurgy (Azevedo, CabreraSerrenho, & Allwood, 2018), and food processing (Gijón-Arreortúa & Tecante, 2015). In contrast with the continuous and transparent flow of fluid, the flow of solid particles is mainly determined by the complex contact force acting between particles, which makes it difficult to elucidate the mixing mechanism. Bladed mixers are most commonly used in particle mixing; however, few studies have investigated the factors that affect the flow and mixing behavior of particles. As a result, the

∗ Corresponding authors. E-mail addresses: [email protected] (Z. Cai), [email protected] (Z. Gao).

design of mixers for particles is usually guided by experience, with little knowledge on the effects of impeller configuration and mixer operating conditions on particle flow and mixing behavior. Experimental studies have traditionally measured the torque and power consumption (Bagster, 1969; Stewart, Bridgwater, Zhou, & Yu, 2001), both of which are helpful for understanding the force needed to move blades through a bed of particles. Position emission particle tracking and particle image velocimetry have been used to study the movement of particles in a mixer. WindowsYule and Parker (2014) discussed the effect of mixer geometry on the dynamical and segregative properties of particles adopting position emission particle tracking and established a relationship between the aspect ratio of a mixer and the degree of segregation. Remy, Glasser, and Khinast (2010) employed particle image velocimetry to study the kinematics of particle flow. However, particle image velocimetry can observe only the external movement of particles in the mixer because solid particles are difficult to penetrate.

https://doi.org/10.1016/j.partic.2019.02.002 1674-2001/© 2019 Published by Elsevier B.V. on behalf of Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences.

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Nomenclature Roman letters C Height from the bottom of the mixer to the bottom of the impeller (mm) Dimensionality of the mixer d D Diameter of the impeller (mm) Ddiff Diffusion coefficient Equivalent coefficient of restitution e∗ E∗ Equivalent Young’s modulus (Pa) Young’s modulus of particle i (Pa) Ei Ej Young’s modulus of particle j (Pa) Deviation of the simulation from the experiment Eri FN Normal forces resulting from the contact of particle ij i with particle j (N) Tangential force resulting from the contact of partiFTij cle i with particle j (N) FnN Normal elastic force (N) Normal damping force (N) FdN FtT Tangential elastic force (N) Tangential damping force (N) FdT g Acceleration due to gravity (N/m) Equivalent shear modulus (Pa) G∗ Gi Shear modulus of particle i (Pa) Shear modulus of particle j (Pa) Gj h1 Height of the collision point from the upper surface of the base (m) h2 Height of the collision point from the upper surface of the base (m) H Height of material filling the mixer (mm) Moment of inertia of particle i (kg m) Ii L Height from the particle feeding plate to the horizontal surface (m) m∗ Equivalent mass (kg) mi Mass of particle i (kg) Mass of particle j (kg) mj Lacey mixing index M N rotational speed (rpm) NA Total number of particles A in the mixture Total number of particles B in the mixture NB NA Number of particles A in the sample NB Number of particles B in the sample Average number of particles A in all samples NA Nc Number of contacts between particles Total number of particles Np Nt Number of time steps p Overall proportion p of particles A in the mixture Proportion of green particles in the simulation pei pI Proportion of particles A in each sample psi Proportion of green particles in the experiment Power consumption of the impeller P R Number of revolutions R∗ Equivalent radius (m) Ri Radius of particle i (m) Radius of particle j (m) Rj S1 Average projection distance of the particle on the surface of the base (m) S2 Average projection distance of the particle on the surface of the base (m) Sn Normal stiffness (N/m) St Tangential stiffness (N/m) Observation time (s) t T Diameter of the cylindrical mixer (mm)

vc

vi

→

Velocity of circumferential direction (m/s) Linear velocity of particle i (m/s)

vrel n

Normal component of relative velocity (m/s)

vrel t vx vy

Tangential component of relative velocity (m/s) Velocity of x direction (m/s) Velocity of y direction (m/s) Total volume of particles in the mixer (m3 ) Width of the impeller (mm) Input work per unit volume (J/m3 ) Coordinates of particles in the x direction (m) Coordinates of particles in the y direction (m) Axial coordinate of the ith particle in the jth time interval (m)

→

V w Winput x y z (i, j)

Greek letters ˇ Equivalent coefficient of restitution Normal overlap (m) ın ıt Tangential overlap (m) Rolling friction coefficient r i Poisson’s ratio of particle i j Poisson’s ratio of particle j i Rolling torque of the particle i (N m) Rolling torque resulting from the contact of particle rij i with particle j (N m) ωi Angular velocity of the particle i (rad/s)

Numerical simulation has great advantages in the research of particle flow and particle mixing because it provides much information that cannot be obtained by experiments, such as the contact force between particles and the velocity of every single particle. Using the discrete element method (DEM), Stewart, Bridgwater, and Parker (2001) simulated a bladed mixer with two flat blades. They found that the frictional characteristics of particles affected velocity profiles and mixing dynamics. Remy, Kightlinger, Saurer, Domagalski, and Glasser (2015) experimentally examined the torque and shear stress in a mixer and found that both increased linearly with the total weight of the bed as predicted by DEM simulations. Boonkanokwong, Remy, Khinast, and Glasser (2016) investigated the effect of the number of blades on particle flow and found that the number of blades affects the velocity of particles and that a two- or three-blade mixer realizes better mixing than a one- or four-blade mixer. Chandratilleke, Yu, Stewart, and Bridgwater (2009) studied the effect of the blade inclination angle and the clearance between the impeller and bottom of the mixer on the particle mixing performance using the DEM and found that mixing is related to the velocity field and the forces acting on the particles. Few studies have evaluated which particle velocity component has the greatest effect on particle mixing in a cylindrical mixer. The relationship between the forces acting on the particles and the mixing performance also needs to be studied. In the DEM, the time step is an important factor determining the accuracy and efficiency of the calculation. The time step is inversely proportional to the square root of Young’s modulus according to a principle of DEM calculations (O’Sullivan & Bray, 2004). In practice, a lower Young’s modulus is usually used to increase the time step and thus shorten the computation time. Alizadeh, Bertrand, and Chaouki (2014) studied the effect of Young’s modulus on DEM simulation and found that the use of a low Young’s modulus led to inaccurate particle velocities near the solid wall in a rotating drum at low coefficients of static friction. Chen, Xiao, Liu, and Shi (2017) used the DEM to study the effect of Young’s modulus on

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the mixing of the bed in a rotating drum and found that there is a critical value (between 0.001E0 and 0.0007E0 , where E0 is the true value of Young’s modulus of the particle) above which the effect becomes negligible. Stewart, Bridgwater, Zhou et al. (2001) and Zhou, Yu, Stewart, and Bridgwater (2004) used a low Young’s modulus E = 2.16 × 106 Pa to simulate a bladed mixer and verified the simulation results at a rotational speed of 19 rpm in experiments using positron emission particle tracking. Young’s modulus affects the collision behavior of particles (Chen et al., 2017), but there has been little research on the value of Young’s modulus needed to accurately calculate particle mixing in a bladed mixer at relatively high rotational speeds. The present study obtained experimentally measured parameters to DEM simulations; i.e., the coefficient of rolling friction, static friction, and restitution. The effects of the impeller configuration (i.e., the blade diameter, number of blades, and inclination angle), rotational speed, and fill level on the mixing behavior were investigated. The present paper is organized as follows. We first detail the DEM simulations. We then experimentally verify the simulation results. We finally analyze the effects of different variables on the particle mixing behavior. Numerical setup

Table 1 Dimensions of the mixer. Diameter of cylindrical mixer, T (mm) Height of the material filling, H (mm) Width of impeller, w (mm) Diameter of impeller, D (mm) Height from bottom of mixer to bottom of impeller, C (mm)

(1)

j

Ii

  dωi Ri × FijT + rij . = dt

(2)



FN d

    is given by rij = −r FijN  Ri ωi . We chose the Hertz–Mindlin (no

slip) model as the soft-sphere contact model in the present study to handle particle collisions. Both the normal force and tangential force have damping components; e.g., the damping coefficient, described by Tsuji, Tanaka, and Ishida (1992), is related to the recovery coefficient. The tangential friction obeys the Coulomb friction law (Cundall & Strack, 1979). The rolling friction is calculated using the contact independent directional constant torque model (Sakaguchi, Ozaki, & Igarashi, 1993). The normal elastic force FnN is given by FN n =

4 ∗ √ ∗ 32 E R ın , 3

(3)

where ın is the normal overlap. The equivalent elastic modulus E ∗ and equivalent radius R∗ are defined by (1 − j2 ) (1 − i2 ) 1 = + , E∗ Ej Ei

(4)

5  ˇ Sn m∗ vrel n , 6 →

= −2

where m∗ =



1 mi

+

1 mj

−1

(6) →

is the equivalent mass and vrel n is the nor-

mal component of the relative velocity. ˇ and Sn (normal stiffness) are defined by



lne∗ ln2 e∗ + 2

Sn = 2E ∗

,

(7)



R∗ ın ,

(8)

e∗

where is the equivalent coefficient of restitution. The tangential elastic force FTt is defined by FTt = −ıt St ,

(9)

where ıt is the tangential overlap and the tangential damping St is defined by St = 8G∗ G∗ =

j

Both Eqs. (1) and (2) are written for all three coordinate directions. Here, mi , vi , Ii , ωi , and Ri are respectively the mass, linear velocity, moment of inertia, angular velocity, and radius of partiT cle i. g is the gravitational acceleration. FN ij and Fij are respectively the normal and tangential forces resulting from the contact of T N T T N particle i with particle j. HereFN ij = Fn + Fd and Fij = Ft + Fd . The effect of rolling friction is included in the torque termrij , which

(5)

where Ei ,  i , and Ri are respectively Young’s modulus, Poisson’s ratio, and the radius of particle i. The normal damping force FN d is given by

Discrete element method

  dv mi i = FijN + FijT + mi g, dt

210 42/63/84/105/126 40 160/180 23

1 1 + , Ri Rj

R∗ =

ˇ=

The DEM was proposed in 1971 on the basis of molecular dynamics. The basic principle of the DEM is to separate discontinuous materials into a set of rigid elements and then make each rigid element satisfy equations of motion. We eventually obtain the complete motion of all discontinuous materials. We used EDEM 2.5.1 DEM software, describing the motion of each particle by

3



R∗ ın ,

2 − i2 Gi

+

(10)

2 − j2 Gj

,

(11) E . Gi and Gj 2(1+) T and j. Fd is defined

where G* is the equivalent shear modulus and G = are respectively the shear moduli of particles i by



FTd

= −2

5  ˇ St m∗ vrel t , 6 →

(12)

→

is the tangential component of the relative velocity. where vrel t The rolling friction is important in the simulation and defined by the torque i applied to the contact surface: i = −r FnN Ri ωi ,

(13)

where r is the coefficient of rolling friction, Ri is the distance from the contact point to the center of mass of the particle, and ωi is the unit angular velocity vector of the particle at the point of contact. Geometry of equipment The size of the mixer is determined by the number of particles used in the simulation. The diameter of the impeller needs to be large enough to ensure that as many particles as possible can be driven by the impeller. A schematic of the cylindrical bladed mixer used in this research is presented in Fig. 1(a) while the side-by-side initial loading is presented in Fig. 1(b). The dimensions of the mixer and their descriptions are given in Table 1. The impeller types are presented in Fig. 2.

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Fig. 1. Schematic of (a) a mixer and (b) side-by-side initial loading.

Fig. 2. Schematics of six different impellers.

Determination of parameters in the DEM Two types of parameters are used in the Hertz–Mindlin (no slip) contact model. (1) Poisson’s ratio, Young’s modulus, and density are intrinsic properties of materials and are usually unaffected by the operating conditions and can be found in the literature. (2) The coefficient of rolling friction, coefficient of static friction, and coefficient of restitution are contact parameters of materials and come into play only when two objects come into contact and are related to both contacting objects. All these parameters vary dramatically with the surface conditions; e.g., friction coefficients may differ greatly if two glass balls are made from the same material but have different surface roughness. It is therefore difficult to obtain the contact parameters from a physical handbook or database. These parameters must be obtained by experimental measurement. Parameter matching is a common experimental method of determining contact parameters in the DEM (e.g., the rest angle and storehouse unloading) by adjusting the contact parameters to make the angle or unloading mass flow rate of particles consistent with the real situation.

Coefficient of restitution In measuring the coefficient of restitution, a common method is to use a high-speed camera to capture the speed of a particle before and after the bounce of the particle (Barrios, de Carvalho, Kwade, & Tavares, 2013). However, this method has strict requirements for experimental equipment. We adopted an ingenious method of measuring the coefficient of restitution in a previous study (Yang, Yang, & Li, 2009). A schematic of the measuring device is presented in Fig. 3. The measuring device consists of a base, adjustment screws, feeding plate, and collision plate. The vertical height of the collision plate can be adjusted using the adjustment screws. The measured particle is dropped from the feeding plate, the height of which is also adjustable. According to the principle of kinematics, we    obtain the coefficient of restitution as Cr = uvnn = ux − uy / 2gL. In this experiment, the particle–particle coefficient of restitution and (particle–steel) coefficient of restitution were respectively 0.624 ± 0.041 and 0.706 ± 0.031.

Coefficients of static and rolling friction The coefficients of static and rolling friction between a particle and steel are obtained from Alian, Ein-Mozaffari, and Upreti (2015). A storehouse unloading experiment is typically conducted

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Table 2 Material properties.

Fig. 3. Schematic of the measuring device. 0. origin of coordinates, 1. base, 2. screw for adjusting height, 3. supporting pole, 4. feeding plate, 5. particle, 6. collision plate.

to measure friction coefficients. We used four equally sized steel plates and a base plate (with all plates made from the same material as the mixer) to form a box having the dimensions of 100 mm × 100 mm × 120 mm. By adding 6140 glass balls (having a total mass of 1 kg) into the box and pulling out one of the steel plates, the balls slide free and form a slope. The coefficients of friction between the particles can be determined by adjusting the friction in the simulation and comparing the simulated angle of the slope with the experimentally measured angle. Simulation and experimental results are shown in Fig. 4. In this comparison experiment, we measured the slope angle directly with a protractor instead of using a camera because the steel plate is not transparent; the angle was 23◦ ± 1◦ . We used binarization and edge detection in the image processing of the simulation (Wang et al., 2016) to obtain the slope angle. We found that the simulation angle was 22.99◦ for a coefficient of the rolling friction of 0.01 and coefficient of static friction of 0.45, which almost matched the experimental result. Material properties based on data taken from the literature and obtained in the above matching experiment are listed in Table 2.

Modeling validation Torque In the simulation, the torque on the geometry is defined as the normal contact force multiplied by the distance from the rotating axis to the contact point. Both the simulated and experimental torque values are the average of instantaneous values. Fig. 5 compares the simulated and experimental values of the torque.

Variable

Value

Coefficient of rolling friction (particle–particle) Coefficient of rolling friction (particle–steel) Coefficient of static friction (particle–particle) Coefficient of static friction (particle–steel) Coefficient of restitution (particle–particle) Coefficient of restitution (particle–steel) Young’s modulus (particle) (Pa) Young’s modulus (steel) (Pa) Particle diameter (random generation) (mm) Poisson’s ratio(particle) Poisson’s ratio (steel) Density (particle) (kg/m3 ) Density (steel) (kg/m3 ) Time step (s)

0.01 0.01 0.45 0.5 0.624 ± 0.041 0.706 ± 0.031 7.2 × 1010 2.04 × 1011 4.75–5.25 0.2 0.285 2480 7930 <10−6

Table 3 Average simulation error for different Young’s moduli. Young’s modulus (Pa)

Average error at different rotational speeds (%)

Average error at different fill levels (%)

7.2 × 1010 108 106

2.019 4.075 17.243

3.732 5.146 17.308

As shown in Table 3, when Young’s modulus (particle) E = 7.2 × 1010 Pa, the maximum error is 5.146%, and the average error in the simulation is 2.019%, indicating that the simulation results are in good agreement with the experimental results. The difference between the experiments and simulations decreases when Young’s modulus becomes closer to the true value for the material. We therefore used the true Young’s modulus to calculate the particle motion and mixing with time permits. Surface particle distribution A top-surface particle distribution experiment was conducted to confirm the simulated particle motion. In this experiment, a twoflat-blade mixer with a diameter of 160 mm was used at a rotational speed of 79 rpm and a fill level H/T of 0.4. White and green particles were made of the same material and had the same size distribution. During the experiment, an image of the top surface was captured at each revolution of the shaft, and the results are compared with the simulation results in the first two columns of Fig. 6. For quantitative comparison of the experimental and simulated top-surface particle distributions, we divided the image into 26 blocks and conducted image analysis to identify the green particles in each block. According to the area of green particles, we obtain the proportion of green particles in each block. The relative of the simulation from the experiment can be expressed deviation as Eri =

(psi −pei )2 , pei

where Eri is the relative deviation, psi is the

Fig. 4. Comparison of experimental and simulated storehouse unloading.

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Fig. 5. Comparisons of simulated and experimental values of torque.

proportion of green particles in the simulation, and pei is the proportion of green particles in the experiment. The right column in Fig. 6 shows Eri in each block; i.e., the deviation of the simulated proportion of green particles from the experimental value. The difference between the simulation and experimental values decreases as the block becomes lighter. It is seen that the simulated particle distributions are in good agreement with the experimental results. Results and discussion

The initial loading in the experiment is side by side and the rotational speed is fixed at 79 rpm. Sections on the degree of mixing and the efficiency of mixing present only the effects of the impeller type on the degree of mixing and the efficiency of mixing. Sections on the circumferential velocity, axial diffusion coefficient, and coordination number compare the circumferential velocity, axial diffusion coefficient, and coordination number of different impeller types and provide possible explanations for the results. Degree of mixing In the simulation, the degree of mixing of particles in the mixer was analyzed using the Lacey mixing index M (Lacey, 1954). The approach is based on the change in relative mixing variance with respect to the maximum mixing variance of a binary mixture, with a value of zero representing complete segregation and a value of 1 representing perfect mixing. The proportion p of particles A in the mixture is NA , NA + NB

(14)

where NA and NB are respectively the numbers of particles A and B in the mixture. The proportion of particles A in each sample is pI =

NA , NA + NB

(15)

where N  A and N  B are respectively the numbers of particles A and B. We have s02 = p(1 − p),

(16)

where S02 is the variance of the initial distribution. 1 (pI − p)2 . n n

s2 =

I=1

where S 2 is the distribution variance at this moment.

sr2 =

(17)

p(1 − p) NA

,

(18)

Where Sr2 is the variance of the completely mixed distribution.where N  A is the average number of particles A in all samples. We have M=

Mixing behavior for different impeller types

p=

Because any two particles in this study are identical,

s02 − s2 s02 − sr2

.

(19)

Cubic cells were used to divide the cylindrical mixer into tens of parts in the calculation of M; the cubic cells are shown in Fig. 7. The size of a cubic cell affected the results of Alian, Ein-Mozaffari, Upreti, and Wu (2015), who studied the effect of the cell size on the Lacey mixing index and found that the mixing index M decreased as the grid became finer. They used cubes having a capacity of about 82 particles each whereas we used cubes with a capacity of 73 particles each in our study. Mathematically, we know that a small number of particles in a cube can change the variance of the proportion of particles A in samples (Chandratilleke, Zhou, Yu, & Bridgwater, 2010). We must therefore delete cells that contain few particles. The optimum cube size depends on the specific situation. We therefore arranged 512 (8 × 8 × 8) cubes in descending order of the number of particles and chose the 400 cubes having the greatest number of particles and discarded the rest, thus keeping 98% of the particles. Fig. 8(a) shows variations in the Lacey mixing index with the number of revolutions for different impeller configurations (i.e., the blade diameter and number of blades). When the degree of mixing reaches 95% of the final degree of mixing, the needed revolutions is called a mixing revolution, and it is 7–9 revolutions in our work (Oldshue, 1983), the same as in the literature (Boonkanokwong et al., 2016; Halidan, Chandratilleke, Chan, Yu, & Bridgwater, 2014). The mixing performance of the 180D two-flat-blade mixer is better than that of the 160D two-flat-blade mixer. The particle mixing mechanisms in a bladed mixer are convection, diffusion, and shearing (Bridgwater, 2012). After the mixer is turned on, the Lacey mixing index increases quickly, then increases at a lower rate, and finally reaches a relatively steady value. As a result, the mixing performance of the 160-mm two-flat-blade mixer is not as good as that of the 180-mm two-flat-blade mixer. At the same rotational speed, the smaller mixer has both a smaller swept area and a lower tip velocity than the larger mixer for weaker convection. When the number of blades is increased from two to four, the final Lacey mixing indices for all flat-blade mixers are higher than 0.98 and the difference between these final Lacey mixing indices is less than 1%. However, the comparison shows that the three-flat-blade mixer provides slightly better mixing than the two- and four-flat-blade

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Fig. 6. Comparison of experimental and simulated surface particle distributions.

mixers. This result does not support the idea that an impeller with more blades provides better mixing. Fig. 8(b) shows the variations in the Lacey mixing index with the number of revolutions for different impeller types (i.e., flat or pitched blades). The final Lacey mixing index can exceed 0.98. Before eight revolutions, the Lacey mixing index of the threeflat-blade mixer is higher than those of the down-pumping and up-pumping mixers for the same number of revolutions. The com-

parison of mixing performance shows that the three-flat-blade mixer provides better mixing than the three-45◦ -blade downpumping and three-45◦ -blade up-pumping mixers.

Efficiency of mixing The input power consumption should be considered in evaluating the mixing performances of different blades (Bao, Lu, Cai, &

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the effect of the impeller type on the Lacey mixing index at a given specific input work. At the same input work, the Lacey mixing indices for the 180D two- and three-blade mixers are almost the same and both are higher than those for the 180D four-flatblade mixer and 160D two-flat-blade mixer. We see that when the mixing index is lower than 0.7, the Lacey index curves for all flatblade mixers almost coincide, indicating that the mixing process is hardly affected by the type of impeller when convection plays the main role in mixing. Fig. 9(b) shows that the three-45◦ -blade up-pumping mixer has better mixing efficiency than the three-flatblade and three-45◦ -blade down-pumping mixers. This conclusion is not the same as that obtained from Fig. 8(b), implying that the three-45◦ -blade up-pumping mixer has lower power consumption than the other two mixers at a given rotational speed.

Fig. 7. Schematic of cubic cells.

Gao, 2018). We propose an equation with which to calculate the specific input work in the mixing: Winput =

60PR , NV

(20)

is the input work per unit volume, J/m3 ; P is the shaft

where Winput power consumption in the mixing, J/s; R is the number of revolutions; N is the rotational speed, rpm; and V is the total volume of the particles in the mixer, m3 . The comparison of the mixing efficiencies (i.e., the mixing performances at a given specific input work) of different mixers is important in industrial design and application. Fig. 9(a) shows

Circumferential velocity In the particle mixing process, the average velocity of particles indicates the ability of particles to move and strongly affects the mixing. Fig. 10 shows that the mixer can be divided into three zones: an above-blade zone, blade zone, and below-blade zone. The particle motions in the mixer are divided into circumferential and axial directions and the circumferential velocity vc requires a simple conversion: vc = vy 

x x 2 + y2

− vx 

y x 2 + y2

,

(21)

where vx , vy , and vc are respectively particle velocities along the x, y, and circumferential directions. The origin is at the center of the mixer base. The average particle circumferential velocity in each zone is shown in Fig. 11. For all bladed mixers, the particle circumferential velocity in the below-blade zone is lower than that in the other

Fig. 8. Effect of the impeller type on the relationship between the Lacey mixing index and number of revolutions.

Fig. 9. Effect of the impeller type on the relationship between the Lacey mixing index and input work.

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or three-45◦ -blade up-pumping mixer. The down-pumping and up-pumping mixers are likely to push the particles up and down, respectively, and the circumferential velocities of these mixers are thus lower than those of a flat-blade mixer. To relate the circumferential velocity to the mixing performance shown in Fig. 8, we note the general consistency except for impeller IV, implying that the circumferential velocity is an important factor but not the only factor affecting the mixing performance. We also note that the particle circumferential velocity in the zone below the impeller is lower than that in other zones, indicating that it is difficult for the blades to drive particles below the impeller. The impeller should therefore be installed as low as possible for best particle mixing near the mixer bottom.

Fig. 10. Schematic of three zones in the mixer.

two zones, indicating that the ability of the particles to move in the below-blade zone is relatively poor. The circumferential velocity of the 180D two-flat-blade mixer is higher than that of the 160D two-flat-blade mixer in each zone. From impeller II to impeller IV, the number of blades increases from two to four and the circumferential velocity in each zone increases but not linearly with the number of blades. The increase in the circumferential velocity decreases with an increasing number of blades. For impellers III, V, and VI, the circumferential velocity of the three-flat-blade mixer is higher than that of either the three-45◦ -blade down-pumping

Axial diffusion coefficient To investigate the ability of particles to move in the axial direction, the DEM, a method commonly applied in molecular dynamics (Chandratilleke, Dong, & Shen, 2018; Ebrahimi, Yaraghi, Ein-Mozaffari, & Lohi, 2018; Keffer, 2001), was used to calculate the axial diffusion coefficient for the mixer. In DEM simulation, the trajectory information of each particle can be collected and the mean-squared displacement (MSD) of the particle in the axial direction is then derived. The MSD is defined by 1  2 = [z (i, j + 1) − z (i, j)] , Np Np Nt −1

[z (t0 + t) − z (t0 )]2i,j

(22)

i=1 j=1

Fig. 11. Effect of the impeller type on circumferential velocity (I: 160D two flat blades, II: 180D two flat blades, III: 180D three flat blades, IV: 180D four flat blades, V: 180D three-45◦ blades up-pumping, VI: 180D three-45◦ blades down-pumping).

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Fig. 12. Effect of the impeller type on the axial diffusion coefficient (I: 160D two flat blades, II: 180D two flat blades, III: 180D three flat blades, IV: 180D four flat blades, V: 180D three-45◦ blades up-pumping, VI: 180D three-45◦ blades down-pumping).

where z (i, j) is the axial coordinate of the ith particle in the jth time interval, Np is the total number of particles, and Nt is the number of time steps. The diffusion coefficient Ddiff is defined by Ddiff =

[z (t0 + t) − z (t0 )]2 1 lim , t 2d t→∞

(23)

where t is the observation time and d is the dimensionality of the system. The numerator is the MSD. Fig. 12 shows the effect of the impeller configuration on the axial diffusion coefficient. The axial diffusion coefficient of the 180D two-flat-blade mixer is larger than that of the 160D twoflat-blade mixer. The axial diffusion coefficient decreases with an increasing number of blades; i.e., the axial movement weakens as more flat blades are used. A comparison of the results for impellers III, V, and VI shows no difference in the axial diffusion coefficient between the three-flat-blade mixer and three-45◦ -blade up-pumping mixer. However, the axial diffusion coefficient of the three-45◦ -blade down-pumping mixer is lower than that of the three-flat-blade mixer and that of the three-45◦ -blade up-pumping mixer. There is no consistent relation between the axial diffusion coefficient and the mixing performance. Because the axial diffusion coefficient of the four-flat-blade mixer is about half that of the three-flat-blade mixer, the four-flat-blade mixer has a higher circumferential velocity but lower mixing performance than the three-flat-blade mixer. The axial diffusion coefficient has a weaker effect than the circumferential velocity on the mixing performance. The mixing performance of the three-flat-blade mixer is better than that of the four-flat-blade mixer because of the interaction of the circumferential velocity and axial diffusion coefficient. Coordination number The coordination number (CN) is defined as the average number of particles in contact with each particle (Chandratilleke, Yu, Bridgwater, & Shinohara, 2012). If a particle is in contact with three particles, for example, CN of the particle is 3. CN is a representation of the compactness of particles. A low compactness means that the gaps between particles are large and the particles are more likely to pass through these gaps for better mixing. CN is defined by CN =

2Nc , Np

(24)

where Nc is the total number of contacts between particles and Np is the total number of particles. Fig. 13 shows CN for the mixers with different blades. CN for the 180D two-flat-blade mixer is higher than that for the 160D two-

Fig. 13. Effect of the impeller type on the coordination number (I: 160D two flat blades, II: 180D two-flat-blades, III: 180D three flat blades, IV: 180D four flat blades, V: 180D three-45◦ blades up-pumping, VI: 180D three-45◦ blades down-pumping).

flat-blade mixer. However, there is no difference in CN between the three-flat-blade mixer and four-flat-blade mixer, but CN for the two-flat-blade mixer is higher than that for the three-flat-blade mixer and that for the four-flat-blade mixer, indicating that the particle layer in the two-flat-blade mixer is relatively compact. CN values for a three-flat-blade mixer and up-pumping mixer are almost the same, indicating that increasing the number of blades hardly increases the space between particles in mixing. CN values for the three-flat-blade mixer and three-45◦ -blade up-pumping mixer are almost the same, and both are smaller than those for the 45◦ -blade down-pumping mixer. A comparison of the Lacey mixing index in Fig. 8 and CN in Fig. 13 shows that the mixing performance is not directly related to CN, implying that particle compactness affects mixing but is not an important factor. Mixing behavior for different operating parameters The rotational speed and the fill level of particles are important parameters of particle mixing in terms of industrial design. To obtain the effects of the rotational speed and fill level on particle mixing, we selected the 160D two-flat-blade mixer and kept the rotational speed constant at 79 rpm or the fill level constant at H/T = 0.3. Degree of mixing Fig. 14(a) shows the variation in the Lacey mixing index with the number of revolutions at different rotational speeds. When the Lacey mixing index is lower than 0.7, the mixing curves are almost the same for different rotational speeds, indicating that the rotational speed hardly affects the Lacey mixing index because convection plays the main role in particle mixing. When the Lacey mixing index is higher than 0.7, the mixing performance increases with the rotational speed because diffusion becomes important and the diffusion rate depends on the tip velocity of the blade. The power consumption increases with the rotational speed, and the comparison of mixing efficiencies at different rotational speeds is discussed in the section on the efficiency of mixing. Fig. 14(b) shows variations in the Lacey mixing index with the number of revolutions at different fill levels. At Lacey mixing indices lower than 0.7, the Lacey mixing index increases with a decrease in the fill level. The final Lacey mixing index at H/T = 0.2 is lowest because most particles are located under the blade and the mixing ability of the particles in the below-blade zone is poor as discussed

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Fig. 14. Effect of (a) the rotational speed and (b) the fill level on the relationship between the Lacey mixing index and number of revolutions.

Fig. 15. Effect of (a) the rotational speed and (b) the fill level on the relationship between the Lacey mixing index and input work.

in the section on the circumferential velocity. The mixing performance at H/T = 0.4 is best and it is recommended for a rotational speed of 79 rpm. Efficiency of mixing Fig. 15(a) shows the relationships between the Lacey mixing index and input work at different rotational speeds. We see that the Lacey mixing indices are almost the same at different rotational speeds except 59 and 79 rpm, an indication that the mixing efficiency is independent of the rotational speed, provided that the rotational speed is not too low. Fig. 15(b) shows the relationships between the Lacey mixing index and input work at different fill levels. A conclusion similar to that drawn in the previous section, on the degree of mixing, can be made in that the mixing efficiency at H/T = 0.4 is best. The mobility of particles in the zone under the blade is low, the proportion of particles in this zone is relatively high at H/T = 0.2 and H/T = 0.3, and it thus takes much energy to reach a higher Lacey mixing index. Similarly, the effect of blades on particles far from the blades is even weaker at H/T = 0.5 and H/T = 0.6. Circumferential velocity Fig. 16 shows the effects of rotational speed and fill level on the circumferential velocity. The circumferential velocity increases with an increase in rotational speed and decreases with an increase in fill level. The relationship of the rotational speed with circumferential velocity is the same as the relationship of the mixing performance mentioned in the section on the degree of mixing in that the circumferential velocity and mixing performance increase with the rotational speed. Fig. 16(b) shows the relationship between the circumferential velocity and mixing performance; there is no general consistency between fill levels and mixing per-

formance but all differences in circumferential velocity are within 10% except for H/T = 0.6. Axial diffusion coefficient Fig. 17(a) shows the effect of rotational speed on the axial diffusion coefficient. The axial diffusion coefficient increases with an increase in rotational speed. This relationship is the same as the relationship for mixing performance mentioned in the section on the degree of mixing. The increase in velocity of the blade tip strengthens particle diffusion and thus particle movements in both the circumferential and axial directions. Fig. 17(b) shows the effect of the fill level on the axial diffusion coefficient. The axial diffusion coefficient is highest at H/T = 0.4. As a result, the mixing efficiency is highest at this H/T value. A comparison of Fig. 14(b) with Fig. 17(b) shows that a higher axial diffusion coefficient provides better mixing performance. We consider two reasons for this relationship; one is that the difference in circumferential velocity is small while the other is that mixing performance is affected by several factors, including the circumferential velocity and axial diffusion coefficient. The axial diffusion coefficient therefore plays an important role in particle mixing when the difference in circumferential velocity is small. Coordination number Fig. 18(a) shows the effect of the rotational speed on the coordination number. CN decreases as the rotational speed increases, an indication that the particles loosen as the rotational speed increases. Fig. 18(b) shows the effect of the fill level on CN. As the fill level increases, CN increases and the particles become compact. A comparison of Figs. 18 and 14 shows that CN is not an important factor of the mixing performance.

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Fig. 16. Effects of (a) the rotational speed and (b) the fill level on the circumferential velocity.

Fig. 17. Effects of (a) the rotational speed and (b) the fill level on the axial diffusion coefficient.

Fig. 18. Effects of (a) the rotational speed and (b) the fill level on the coordination number.

Conclusions We studied the effects of the impeller type (i.e., blade diameter, inclination angle, and number), rotational speed, and fill level on the mixing ability, flow, and compactness of particles in a bladed mixer. Simulation results of the torque and surface particle distribution were validated in experiments. The Lacey mixing index, input work per unit volume, circumferential velocity, axial diffusion coefficient, and coordination number for different impellers and operating parameters were compared and the following conclusions drawn. (a) Effects of the impeller type (i.e., blade diameter, inclination angle, and number): When the diameter of the impeller increases, the Lacey mixing index, mixing efficiency, circumferential velocity, and axial diffusion coefficient increase whereas the coordination number decreases. The mixing performance and efficiency of the

three-flat-blade mixer are better than those of the two- and fourflat-blade mixers. Except for the three- and four-flat-blade mixers, there is a consistent relationship of the circumferential velocity with the mixing performance of different impellers. The circumferential velocities for both the three-45◦ -blade down-pumping and three-45◦ -blade up-pumping mixers are lower than that for the three-flat-blade mixer. The axial diffusion coefficient of the three-45◦ -bladed down-pumping mixer is smaller than those of the three-flat-blade mixer and three-45◦ -bladed up-pumping mixer. There is no significant difference in the axial diffusion coefficient between the three-flat-blade mixer and three-45◦ -bladed up-pumping mixer, both being larger than that of the three-45◦ bladed down-pumping mixer. A comparison of the relationship of the circumferential velocity, axial diffusion coefficient, and CN with mixing performance reveals that the circumferential velocity has a

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stronger effect on the mixing performance than the axial diffusion coefficient and coordination number. (b) As the rotational speed increases, the mixing performance increases but the mixing efficiency is almost unchanged. Movability in the circumferential and axial directions also increases as the rotational speed increases and the particles become loose. The mixing performance and efficiency at a fill level of H/T = 0.4 is best because the movability of particles far from the blades is low. The circumferential velocity decreases with an increasing fill level. The axial diffusion coefficient at H/T = 0.4 is largest that at H/T = 0.2 and that at H/T = 0.6 are smallest. When the fill level increases, CN increases, and the particles become compacted. Our study provides an entire methodology for simulating the mixing of particles. Our future work will include (1) the effect of the blade rotation frequency on the mixing behavior at different fill levels and (2) a method of simulating the cohesive mixing of particles. Acknowledgement The authors gratefully acknowledge financial support from the Scientific Research and Technology Development Projects of the China National Petroleum Corporation (No. 2016B-2605). References Alian, M., Ein-Mozaffari, F., & Upreti, S. R. (2015). Analysis of the mixing of solid particles in a plowshare mixer via discrete element method (DEM). Powder Technology, 274, 77–87. Alian, M., Ein-Mozaffari, F., Upreti, S. R., & Wu, J. (2015). Using discrete element method to analyze the mixing of the solid particles in a slant cone mixer. Chemical Engineering Research & Design, 93, 318–329. Alizadeh, E., Bertrand, F., & Chaouki, J. (2014). Comparison of DEM results and Lagrangian experimental data for the flow and mixing of granules in a rotating drum. AIChE Journal, 60, 60–75. Azevedo, J. M. C., CabreraSerrenho, A. C., & Allwood, J. M. (2018). Energy and material efficiency of steel powder metallurgy. Powder Technology, 328, 60–75. Bagster, D. F. (1969). The prediction of the force needed to move blades through a bed of cohesionless granules. Powder Technology, 3, 153–162. Bao, Y., Lu, Y., Cai, Z., & Gao, Z. (2018). Effects of rotational speed and fill level on particle mixing in a stirred tank with different impellers. Chinese Journal of Chemical Engineering, 26, 1383–1391. Barrios, G. K. P., de Carvalho, R. M., Kwade, A., & Tavares, L. M. (2013). Contact parameter estimation for DEM simulation of iron ore pellet handling. Powder Technology, 248, 84–93. Boonkanokwong, V., Remy, B., Khinast, J. G., & Glasser, B. J. (2016). The effect of the number of impeller blades on granular flow in a bladed mixer. Powder Technology, 302, 333–349. Bridgwater, J. (2012). Mixing of powders and granular materials by mechanical means—A perspective. Particuology, 10, 397–427. Cavinato, M., Bresciani, M., Machin, M., Bellazzi, G., Canu, P., & Santomaso, A. C. (2010). Formulation design for optimal high-shear wet granulation using on-line torque measurements. International Journal of Pharmaceutics, 387(1-2), 48–55. Chandratilleke, G. R., Dong, K. J., & Shen, Y. S. (2018). DEM study of the effect of bladesupport spokes on mixing performance in a ribbon mixer. Powder Technology, 326, 123–136.

13

Chandratilleke, G. R., Yu, A. B., Bridgwater, J., & Shinohara, K. (2012). A particlescale index in the quantification of mixing of particles. AIChE Journal, 58, 1099–1118. Chandratilleke, G. R., Yu, A. B., Stewart, R. L., & Bridgwater, J. (2009). Effects of blade rake angle and gap on particle mixing in a cylindrical mixer. Powder Technology, 193, 303–311. Chandratilleke, G. R., Zhou, Y. C., Yu, A. B., & Bridgwater, J. (2010). Effect of blade speed on granular flow and mixing in a cylindrical mixer. Industrial & Engineering Chemistry Research, 49, 5467–5478. Chen, H., Xiao, Y. G., Liu, Y. L., & Shi, Y. S. (2017). Effect of Young’s modulus on DEM results regarding transverse mixing of particles within a rotating drum. Powder Technology, 318, 507–517. Cundall, P. A., & Strack, O. D. L. (1979). A discrete numerical model for granular assemblies. Géotechnique, 29(1), 47–65. Ebrahimi, M., Yaraghi, A., Ein-Mozaffari, F., & Lohi, A. (2018). The effect of impeller configurations on particle mixing in an agitated paddle mixer. Powder Technology, 332, 158–170. Gijón-Arreortúa, I., & Tecante, A. (2015). Mixing time and power consumption during blending of cohesive food powders with a horizontal helical double-ribbon impeller. Journal of Food Engineering, 149, 144–152. Halidan, M., Chandratilleke, G. R., Chan, S. L. I., Yu, A. B., & Bridgwater, J. (2014). Prediction of the mixing behaviour of binary mixtures of particles in a bladed mixer. Chemical Engineering Science, 120, 37–48. Keffer, D. (2001). The working man’s guide to obtaining self diffusion coefficients from molecular dynamics simulations. Knoxville, USA: Department of Chemical Engineering, University of Tennessee. Lacey, P. M. C. (1954). Developments in the theory of particle mixing. Journal of Applied Chemistry, 4, 257–268. Oldshue, J. Y. (1983). Fluid mixing technology. New York: Mcgraw-Hill. O’Sullivan, C., & Bray, J. D. (2004). Selecting a suitable time step for discrete element simulations that use the central difference time integration scheme. Engineering Computations, 21, 278–303. Remy, B., Canty, T. M., Khinast, J. G., & Glasser, B. J. (2010). Experiments and simulations of cohesionless particles with varying roughness in a bladed mixer. Chemical Engineering Science, 65, 4557–4571. Remy, B., Glasser, B. J., & Khinast, J. G. (2010). The effect of mixer properties and fill level on granular flow in a bladed mixer. AIChE Journal, 56, 336–353. Remy, B., Kightlinger, W., Saurer, E. M., Domagalski, N., & Glasser, B. J. (2015). Scale-up of agitated drying: Effect of shear stress and hydrostatic pressure on active pharmaceutical ingredient powder properties. AIChE Journal, 61, 407–418. Sakaguchi, H., Ozaki, E., & Igarashi, T. (1993). Plugging of the flow of granular materials during the discharge from a silo. International Journal of Modern Physics B, 7, 1949–1963. Stewart, R. L., Bridgwater, J., & Parker, D. J. (2001). Granular flow over a flat-bladed stirrer. Chemical Engineering Science, 56, 4257–4271. Stewart, R. L., Bridgwater, J., Zhou, Y. C., & Yu, A. B. (2001). Simulated and measured flow of granules in a bladed mixer—A detailed comparison. Chemical Engineering Science, 56, 5457–5471. Tsuji, Y., Tanaka, T., & Ishida, T. (1992). Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technology, 71, 239–250. Wang, Y., Liang, Z., Zhang, D., Chui, T., Shi, S., Li, K., et al. (2016). Calibration method of contact characteristic parameters for corn seeds based on EDEM. Transactions of the Chinese Society of Agricultural Engineering, 32(22), 36–42. Windows-Yule, C. R. K., & Parker, D. J. (2014). Self-diffusion, local clustering and global segregation in binary granular systems: The role of system geometry. Powder Technology, 261, 133–142. Yang, M., Yang, L., & Li, Q. (2009). Simple measurement of restitution coefficient of granular material and its application. Journal of Agricultural Mechanization Research, 31, 25–27. Zhou, Y. C., Yu, A. B., Stewart, R. L., & Bridgwater, J. (2004). Microdynamic analysis of the particle flow in a cylindrical bladed mixer. Chemical Engineering Science, 59, 1343–1364.

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