Experimental and DEM studies on the particle mixing performance in rotating drums: Effect of area ratio

Powder Technology 314 (2017) 182–194

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Experimental and DEM studies on the particle mixing performance in rotating drums: Effect of area ratio Xiangwu Xiao a, Yuanqiang Tan b,⁎, Hao Zhang c, Rong Deng a, Shengqiang Jiang a a b c

School of Mechanical Engineering, Xiangtan University, Hunan 411105, China Institute of Manufacturing Engineering, Huaqiao University, Xiamen 361021, China School of Energy and Environment, Southeast University, Nanjing 210096, China

a r t i c l e

i n f o

Article history: Received 27 July 2016 Received in revised form 10 December 2016 Accepted 16 January 2017 Available online 19 January 2017 Keywords: Discrete element method Mixing performance Area ratio Rotating drum

a b s t r a c t The mixing processes of equal-sized acrylonitrile butadiene Styrene copolymers beads in a rotating drum were simulated using the Discrete Element Method (DEM) at different rotational speeds and filling degrees. The contacting parameters of the DEM models were determined by a series of experiments including high rebound test, friction and wear test, stacking angle test and L-box test. The validity of the DEM simulations was confirmed by comparing the numerical results with the experimental ones. Quantitative criteria of particles mixing performance which is characterized by the mixing degree and mixing time were proposed based on the mathematical statistical analysis of simulation results. A new parameter C which is defined as the ratio of the active region area to the whole bed area was suggested to describe the active region in the mixing process. Our numerical results reveal that the parameter C has a significant influence on the mixing time but little influence on the mixing degree. With the increase of the rotational speed and the decrease of the filling degree, the parameter C increases but the mixing time decreases. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The products in the industries spanning from pharmaceuticals, minerals, agriculture and other fields, increasingly depend on the reliable granular flow and uniform granular mixing [1,2]. The rotating drum is commonly used for the mixing process of granular materials, and a vast number of researches have been devoted to the investigation of particle mixing performance in rotating drums during the past decades [3–5]. However, the particle mixing performance is still puzzling even qualitatively due to the lack of proper method to characterize the stochastic nature of the particles. Considerable efforts have been made in the past to quantize the particle mixing performance. The use of the variance of sample concentrations was perceived to be a sensible way ahead with the unmixed state and the randomly mixed state providing reference conditions [4,5]. There were two general methods available for sampling to measure the particle mixtures, one was sampling with off-line assessment in which material was withdrawn and analyzed. The other was on-line assessment in which particulate material was remained in the process [6]. The former type usually used a thief sampler [1,7], which raised a number of troubles such as the thief sampler was too slow in use, the displacement of material was induced by the insertion of the thief ⁎ Corresponding author. E-mail address: [email protected] (Y. Tan).

http://dx.doi.org/10.1016/j.powtec.2017.01.044 0032-5910/© 2017 Elsevier B.V. All rights reserved.

sampler and the measurement results were not continuous. On the contrary, the latter one, on-line assessment, could be carried out by observations of a free surface or through a transparent wall in an external surface or through a transparent wall. This type could be divided into in situ measurements and non-invasive analysis. In situ measurements, the content of powder mixtures could be obtained by the use of a fiberoptic reflection probe [8,9]. But this approach has not been widely used. Non-invasive analysis could be obtained by means of image analysis [10,11]. But there was concern that the use of image analysis at free surfaces or through a transparent wall in an external surface restricted the value, especially in the systems prone to segregation. The measurement of mixing performance is not easy work which is also expansive both in time and money. Recently, high performance computers enable us to simulate precisely the mixing behavior and monitor the mixing performance of particles globally and continuously [12]. Especially, Lagrange methods like the Discrete Element Method (DEM) are widely used to analyze the mixing performance [13,14]. The DEM is originated by Cundall and Strack [15] which can predict the behavior of particles by calculating the motion of every single particle and has been widely adopted as a powerful numerical tool to analyze various engineering issues [16–20]. There are also some numerical simulation works on the mixing processes in rotating drum. It has been revealed that the particle mixing performance in a rotating drum is in relation to some variables, such as particle size, particle density, frictional coefficient, filling degree and rotational speed [21–25].

X. Xiao et al. / Powder Technology 314 (2017) 182–194

In addition, the mixing performance is closely related to the motion mode of rotating drum [14,26]. Six modes of solid motion have been identified in a rotating drum, such as slipping, slumping, rolling, cascading, cataracting and centrifuging [27–29], with the increasing of rotation speed. Industrial-scale rotating drum mixers are usually operated in the rolling mode since it provides superior particle mixing [30, 31]. In the rolling mode, particle bed can be briefly divided into two regions, namely the active region and the static region. Since the mixing behavior mainly occurs in the active region [31], particle motion in this region is crucial to evaluate the mixing performance of the whole drum. Many studies have been carried out to investigate the thickness of the active region by combining theoretical model and experiment results of particles velocity field in rotating drum [32,33]. The particles velocity field was elucidated by using experiments techniques such as magnetic resonance imaging (MRI) [34], positron emission particle tracking (PEPT) [31], particle image velocimetry (PIV) [35] and radioactive particle tracking (RPT) [36]. However, it may be impossible to accurately obtain the particle velocity field in the experiment. The DEM simulations can conquer these drawbacks and provide much more information than the experimental [37]. Therefore, combining DEM simulation results of particles velocity field and theoretical model to acquire the thickness of the active region is feasible though limited available results in this researching field have been reported. Moreover, only using the thickness to describe the active region is not sufficient when there are significant differences on the chord length of particle bed which can be affected by filling degree. Therefore, it is necessary to quantize the area to describe the size of active region. To the best knowledge of the authors, no study has focused on exploring the relationship between the active regions and mixing performance. The aim of the current study is to fill this gap. In this study, DEM was used to simulate the mixing process of equalsized Acrylonitrile butadiene Styrene copolymers (ABS) beads in a rotating drum at different operating conditions including rotational speed and filling degree. The Hertz-Mindlin model was adopted as the interaction rule between the elements. At the same time, a series of experiments such as high rebound test, friction and wear test, stacking angle test and L-box test were designed to determine the contact model parameters of the coefficient of restitution, static friction and rolling friction. The validity of the DEM simulation was confirmed by comparing with the experimental results. Then, a mathematical statistical method was employed to quantitatively measure the particle mixing performance including mixing degree and mixing time. The simulation results of particles velocity field were analyzed to obtain the thickness of the active region and the active region area. A new parameter C was proposed and defined as the ratio of the active region area to the whole bed area. At last, the relationship between the active region and mixing performance were explored. 2. Simulation methods 2.1. Discrete element method Particles are usually assumed as rigid bodies in the DEM, and the motion of each particle is governed by the Newton's second law. Small overlap is allowed at the contact between two particles and between a particle and a wall. A force-displacement law, i.e., constitutive law, is applied at the contact to account for the particle-particle or particle-wall interaction. When a direct contact or collision takes place, the contact force is calculated and an iterative method is used to update the position of the particles [38]. The governing equations of particle motion are described as:

mi

2! Ni !ij !i ∂ ri ¼∑ F þ Fe 2 ∂t j¼1

ð1Þ

Ji

2! Ni ∂ θi !ij !ij !i ¼ ∑ q  F þ K e: 2 ∂t j¼1

183

ð2Þ

Where mi is the mass of the particle i, Ji is the rotational inertia of the ! ! particle i, r i is the position vector of the particle i, θ i is the angle vector ij ! !ij of the particle i, F is the acting force of particle j to particle i, q is the !i arm of force from the acting force to the center of the particle i, F e is the !i external force, K e is the external torque of the particle i, and Ni is the number of particle contacting with the particle i. In this study, the Hertz–Mindlin no-slip model was adopted to calculate the interaction force between the elements including colliding particles and walls [37]. The Hertz–Mindlin model is combined the classical Hertz theory [39] in the normal direction and the Mindlin and Deresiewicz theory [40] in the tangential direction. The calculation of force in this model is given out in Appendix A. 2.2. Determination of discrete element parameters The appropriate choice of discrete element parameters has an important influence on the accuracy and precision of the DEM simulation results. There are two kinds of parameters in Hertz-Mindlin no-slip model: material property parameters and contact model parameters. 2.2.1. Material property parameters In this study, the material property parameters include radius, density, shear modulus and Poisson's ratio. The rotating drum is made of Q235 and the length is 60 mm and the visual cover is made of Polymethylmethacrylate (PMMA) which is shown in Fig. 1. Radius and density were obtained by a series of simple experiments like sieving test and drainage test. The material property parameters are given out in Table 1. 2.2.2. Contact model parameters The contact model parameters involve the coefficient of restitution, static friction and rolling friction. A series of experiments such as high rebound test, friction and wear test, stacking angle test and L-box test were designed to measure the contact model parameters. Collision coefficient of restitution is defined as: e¼

v2 −v1 ¼ v10 −v20

rffiffiffiffiffi h0 H

ð3Þ

where v10 and v20 are the velocity of two objects before the collision, v1 and v2 are the velocity of two objects after the collision, h0 is the height of rebounding of particle and H is the initial height of particle. Particle fell from a specified height and collided with the cover plate, Q235 plate of rotating drum or ABS beads which are shown in Fig. 2. High speed camera Phantom V10 was used to record the experiment process. The measure results of the restitution coefficient of particle-cove, particle-drum and particle-particle are 0.850 ± 0.014, 0.918 ± 0.017 and 0.873 ± 0.024, respectively. The static friction coefficient of particle-drum and particle-cover were measured by friction and wear testing machine UMT-3 which is shown in Fig. 3(a). The angle measured in stacking angle test is usually called dynamic angle of repose, which is a macroscopic friction angle. It is slightly larger than particle contact friction when the particle friction coefficient less than 0.4 [41]. In this paper, stacking angle test was adopted to measure static friction coefficient of particle-particle which is defined as: μ spp ¼

hs r

ð4Þ

where r is the diameter of platform and keeps a constant value of 75 mm, hs is the height of particles stacking. The measurement method is given

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bulkhead

(a)

(b)

Fig. 1. (a) The rotating drum in (a) laboratory; (b) DEM model. Note that the length of rotating drum is adjustable by fixing bulkhead in experiment.

out in Fig. 3(b). The results of the static friction coefficient of particledrum, particle-cover and particle-particle are 0.382 ± 0.015, 0.422 ± 0.041 and 0.360 ± 0.039, respectively. In this study, the coefficient of rolling friction was obtained by combing experiments and simulations: a large number of numerical simulations with changing parameters were conducted to approach the actual experiments. L-box test was applied to ascertain the coefficient of rolling friction. The L-box is composed of Q235 wall and PMMA wall. The results of simulation and experiment are shown in Fig. 4 that the rolling friction coefficient of particle-particle is 0.030 ± 0.011, particle-wall (Q235) is 0.021 ± 0.018 and particle-wall (PMMA) is 0.015 ± 0.014. From above, the contact model parameters were measured and presented in Table 2. 2.3. Simulation conditions The mixing processes of equal-sized ABS beads with two kinds of color in a rotating drum were simulated. The diameter of particles is 4 mm, the density of particles is 1100 kg/m3 and the number of two kinds ABS beads is equal. Simulation conditions are summarized in Table 3. As mentioned above, there are six modes of solids motion including slipping, slumping, rolling, cascading, cataracting, and centrifuging in a rotating drum. The ranges for Froude number, filling degree and critical wall friction coefficient are specified to delimit the types of bed motion. Generally, a characteristic criterion for the motion of particles in rotating drum is the Froude number Fr which is defined by Fr ¼

ω2 R g

ð5Þ

where g is the acceleration due to gravity, R is the radius of rotating drum and ω is the rotational speed of drum. As pointed out by Mellman [29], the critical particle-wall friction coefficient is given by μ spw;c ¼

2sin3 ε sinθ 3πf ð1 þ F r Þ

ð6Þ

where ε is the half bed angle of the circular segment occupied with particles and θ is the dynamic angle of repose which are shown in Fig. 5. f is

Table 1 The material property parameters in simulations. Material properties

Particle (ABS)

Drum (Q235)

Cover (PMMA)

Radius (mm) Density (kg/m3) Shear modulus (Pa) Poisson's ratio

2 1100 2e + 7 0.394

147 7850 2e + 11 0.260

147 1190 3e + 9 0.250

the filling degree which defined as the ratio of the whole bed area Ab to 2

the cross section area of drum Ad where Ad = πR2 and Ab ¼ επR 180 −Lh. It is known that h =R cos ε , L = Rsin ε. So, the filling degree can be expressed as f ¼

ε sinε cosε − : 180 π

ð7Þ

ε can be obtained theoretically from Eq. (7) when f is given out. As industrial-scale rotating drum is usually operated in the rolling mode since it provides superior particle mixing, 10−4 b Fr b 10− 2, f N 0.1 and μspw N μspw , c should be conform. μspw is the static friction coefficient of particle-wall (drum). From Table 2, we know that μspw = 0.382. In order to produce the rolling mode, μspw,c should be less than 0.382. Maximum value of μspw , c is obtained about 0.357 with the current simulation conditions of f = 0.2, ε = 60.537∘, θ = 30.8∘ and Fr = 0.0041. 3. Results and discussions 3.1. Validation of the DEM model Rotating drum tests were performed to show the agreement between the DEM simulations and actual mixing processes. The laboratory test apparatus consists of a rotating drum which was filled with equalsized ABS beads, supported on 8 ball bearings and connected to a servomotor as shown in Fig. 1. In the rotating drum test 1, equal-sized ABS beads with two kinds of color were mixed. The particle mixing process was recorded by a digital camera. Particle mixing states for three cross-sectional views of a rotating drum mixer were shown in Fig. 6. It can be clearly seen that the simulation and experiment results are in qualitative agreement in terms of the spatial distribution of particles. Furthermore, single tracer particle with white color was mixed with red ABS beads in rotating drum test 2 which is shown in Fig. 7(a). Image analysis was adopted to analyze the velocity of single particle in laboratory test. The video recordings of the experiment were decoded into a sequence of pictures and the sequence was converted to gray scale, and pixels belonging to each particle type (white and red) were detected based on their color value. The location information of tracer particle was collected in simulation and experiment which are given out in Fig. 7(b). The u is defined as the mean velocity of tracer particle in x direction and the v is defined as the mean velocity of tracer particle in y direction u¼−

v¼

 0  Δx Δy0 cosθ þ sinθ Δt Δt

Δx0 Δy0 sinθ− cosθ Δt Δt

ð8Þ

ð9Þ

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Fig. 2. Measurement of restitution coefficient of (a) particle-drum; (b) particle-cover; (c) particle-particle.

where Δt = 0.1 s. In this study, we assume that the moment of the Fig. 7(a) is the initial moment. Δx' and Δy' are the displacements of tracer particle in horizontal direction and vertical direction during 0.1 s, respectively. The dynamic angle of repose θ was measured by image analysis which is equal in simulation and experiment. Simulation results and experimental results of single tracer particle mean velocity are displayed in Fig. 8. Tracer particle in active region falls after it detaching from the wall during the time of 0 to 1.2 s and it moves along the rotating drum wall in static region about from 1.2 to 6.2 s. The velocity of tracer particle in streamwise direction u is positive when the particle entrance of the static region and it increased in the falling process. When the particle approaches the drum wall, u decreased, which due to the appearance of congestion phenomenon that u in static region is much smaller than it in the region active region. When the particle enters the static region, v reduced, and it is about zero when particles approaches the straight line consisting of point A and point B which is shown in Fig. 5. v increases after that the tracer particle moves away that line. It is found that the simulation results agree with experimental results of particle velocity. From the rotating drum tests, we can see that the numerical simulation results are in good agreement with the experimental results. It indicates that the DEM model parameters are accurate and the DEM simulation results are reasonable and feasible, especially, the DEM simulation results of particles velocity field.

3.2. Characterization of the mixing performance A crucial issue in the particle mixing is the evaluation of the mixing performance. To ensure the homogeneity of the final product,

characterization of such systems during the mixing process plays an important role. So far, mixing degree of particles was used to characterize the mixing performance and some different mixing indexes such as segregation index, Lacey index, Poole index, mixing indexes using standard deviation, and so on were proposed by various authors to measure the mixing degree of particles [42–45]. However, the mixing performance is not only characterized by the mixing degree, but also characterized by the mixing time. The less mixing time means that the huge economic effects will achieve in engineering applications. In order to assess the mixing performance efficiency of the rotating drum mixer, a mathematical statistical method using standard deviation was utilized in the previous work [2]. Φi is defined as the concentration of red particles in cell i and Φ is the ideal concentration of red particles in the rotating drum: Φi ¼

Φ¼

ai ai þ bi

ð10Þ

N ∑i¼1 ai N ∑i¼1 ðai þ bi Þ

ð11Þ

where ai and bi are the number of red particles and white particles in cell i, respectively. N is the number of uniformly distributed cells within the rotating drum. From this, the mixing index S is defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 N  ∑ X −X S¼ N−1 i¼1 i where X i ¼ ΦΦi ; and X ¼ ferent time.

ð12Þ N

∑i¼1 X i N

: We can obtain the mixing index S at dif-

Fig. 3. (a) The friction and wear testing machine UMT-3; (b) the measurement method for static friction coefficient between particles.

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Fig. 4. The L-Box flow test (top) experimental (bottom) numerical results.

During the mixing process, the mixing and segregation phenomena change dynamically. There are three stages in the particle mixing process as shown in Fig. 9(a), namely the convective mixing stage, the convective and shear interaction stage and the diffuse mixing stage [46]. In the convective mixing stage, the particle mixing speed is high because the mixing process in this stage is mainly on the macro-scale level. In this stage, the convective circulation movement plays a leading role, the diffusion and segregation phenomena are not obvious. However, in the convective and shear interaction stage, the mixing speed of the particles is slowed down due to the fact that the diffusion movement is accelerated. The effects of the convective circulation and the diffusion movements on the mixing process tend to be equal. After some time, the diffusion movement plays a dominant role. Finally, in the diffuse mixing stage, mixing and separation are balanced and the mixing index is fluctuated. The simulation results of mixing index S are given out in Fig. 9(b) at case 3. It can be seen that the numerical results have the same trend with the results of German [46]. In order to obtain the mixing degree and mixing time quantitatively, the polynomial fitting of mixing index S was adopted which is expressed as S ¼ g ðt Þ:

ð13Þ

The mixing index S is a function of time t. We define that mixing process has entered the diffuse mixing stage when the slope of fitting curve equals zero which means that t ¼ t1 ;

dg ðt Þ ¼0 dt

ð14Þ

Linear fitting of mixing index S was adopted during the time period of t1 to end, and it is expressed as S1 ¼ g 1 ðt Þ:

ð15Þ

It is found that the fitting straight line approximate horizontally which means that the mixing process really reaches the diffuse mixing stage, and proves that the mathematical statistical method is correct.

Table 2 The contact model parameters in simulations. Contact properties

Particle-particle

Particle-wall (Q235)

Particle-wall (PMMA)

Coefficient of restitution Coefficient of static friction Coefficient of rolling friction

0.873 0.360 0.030

0.918 0.382 0.021

0.850 0.422 0.015

The first intersection of fitting curve and straight line is very important. We define mixing degree Sa as the value of mixing index that particles reach the uniform state and mixing time Tm as the value of time that particles reach the uniform state: t ¼ T m ; g ðt Þ ¼ g 1 ðt Þ

ð16Þ

Sa ¼ g1 ðT m Þ:

ð17Þ

The lower the value of Sa is, the higher the mixture quality is. We obtained the mixing index S, mixing degree Sa and mixing time Tm of case 3 and which are shown in Fig. 9. In case 3, the mixing degree Sa equals 0.1339 and the mixing time Tm is 93.7485 s. 3.3. Calculation of the area ratio It can be seen that the particle bed can be briefly divided into two regions: a static region and an active region in Fig. 5. The particles in static region motive along the rotating drum wall, and particles in active region will fall after them detaching from the wall. Below the active region is the static region, where the particles are compacted and collectively rotate as a solid body. Since the mixing behavior mainly occurs in the active region, the particle motion in this region is of prime importance for the mixing performance of the whole drum. So, it is necessary to investigate the effect of active region on particles mixing performance in rotating drum. Usually, the active region is characterized by its thickness [31–33]. However, we find that it is insufficient to describe the size of active region using its thickness alone when the chord length of particle bed is mutative which is discussed in Section 3.4. Thus, we calculate the area of active region and propose a new parameter C which is defined as the ratio of the active region area to the whole bed area to characterize the active region. Table 3 Simulation conditions. Case no.

Rotational speed (rpm)

Filling degree

Fr

ε

Forms of transverse motion

1 2 3 4 5 6 7

3 4 5 6 5 5 5

0.40 0.40 0.40 0.40 0.20 0.30 0.45

0.0019 0.0026 0.0041 0.0059 0.0041 0.0041 0.0041

80.925o 80.925o 80.925o 80.925o 60.537° 71.356o 85.491o

ROLLING ROLLING ROLLING ROLLING ROLLING ROLLING ROLLING

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by investigate the picture, every simulation presents a qualitatively similar velocity field. In order to make an efficient comparison of two regions and obtain the area of active region, it is very important to analyze the stream wise of x-direction velocity, transverse of ydirection velocity and the bed surface velocity. Fig. 11 shows the plots of x-direction velocity profile and y-direction velocity profile against y-direction position for case 3. These simulation results are in good agreement with the PEPT data reported by Ding and Dubé [31,36] for the rotating drum. It can be seen that, in a given section normal to the bed surface like x/2 L = 0.3, the maximum x-direction velocity in the active region occurs in the bed surface. With increasing ydirection position, the particle velocity approaches the drum wall velocity in the x-direction. It indicates that the slippage between the wall and particle is negligible. For all sections, particles rotate as a solid body in the static region for δ(x) ≤ y ≤ (R− h), the x-direction velocity ustatic and y-direction velocity vstatic follow:

Fig. 5. Schematic diagram of a rotating drum operated in rolling mode.

3.3.1. Velocity profiles The computation of the thickness of the active region and the area ratio of the active region to the whole bed essentially relies on the particles velocity field. Fig. 10 displays the particles velocity fields of a cross section close to the cover for the case 3. A geometry bin close to the cover was used to draw particles. Particle velocity was analyzed when the gravity center of the particle belonged into the geometry bin. It means that some places of velocity field are vacant. The phenomenon of particles flowing downward from the apex to base in active region and particles being carried upward in static region can be appreciated

ustatic ðx; yÞ ¼ −ωðh þ yÞ

ð18Þ

vstatic ðx; yÞ ¼ ωðx−LÞ:

ð19Þ

h and L can be calculated when f is given out. We can see that the simulation results of ustatic and vstatic are in good agreement with the theoretically results which calculated by Eqs. (18), (19) for the case 3. Fig. 12 gives out x-direction velocity of particles in the bed surface. The bed surface velocity u0 at (x = 0, y = 0) and (x = 2 L, y = 0) approximately equal zero. The maximum bed surface velocity appears in (x = L, y = 0). u0 is approximately symmetrical with respect to the mid-chord position. The symmetric profile has been observed previously by experimental work [31,47]. Polynomial fitting is adopted to describe simulation results of bed surface velocity u0: u0 ¼ f ðxÞ

ð20Þ

Fig. 6. Cross-sectional views of rotating drum mixer (top) experimental (bottom) numerical results. Note that the particle size is 10 mm, rotational speed is 10 rpm, the filling degree is 0.33 and the number of two kinds ABS beads is equal.

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Fig. 7. (a) Location of single tracer particle in experiment and simulation; (b) the trajectory of tracer particle in experiment and simulation during some time. Note that the particle size is 4 mm, rotational speed is 5 rpm and the filling degree is 0.4.

where f(x) can be determined by the simulation results, but it is different for all case. 3.3.2. The area ratio In this study, the area ratio C is defined as C¼

Aa : Ab

ð21Þ

Where Aa is the active region area. In order to calculate Aa, it is necessary to calculate the thickness of the active region δ. Several theoretical models are available to calculate the thickness of active region [31–33]. Ding et al. [31,48] proposed a model based on mass and momentum balances to predict the thickness of the active region. In this model, the thickness of the active region is expressed as: 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3     x2 2 2 2 4 δðxÞ ¼ 2 3Λ þ 1 Lx− þ 4h −2h5: 6 1−Λ 2 3Λ þ 1 1

Where Λ ¼

qffiffiffiffiffiffiffiffiffiffiffi u0 ωhþu0

and it defined as the ratio of α(x) to δ(x). α(x) is the

distance between line of zero velocity and bed surface and u0 is the streamwise velocity along the bed surface. The thickness of the active region is a function of x , L , h , ω and u0. L and h are known when f and R are given out. So the thickness of the active region δ is determined by parameters bed surface velocity u0 and x-direction position. The x-direction velocity of particles in the bed surface u0 can be obtained by the polynomial fitting of simulation results and it is a function of x. So, the thickness of active region δ is just a function of parameter x. With known the function δ(x), the active region area can be calculated through integration as 2L

Aa ¼ ∫ δðxÞdx: 0

ð23Þ

The whole bed area Ab is ð22Þ

Ab ¼ fπR2 :

Fig. 8. The experiment and simulation results of tracer particle mean velocity in (a) x-direction, (b) y-direction.

ð24Þ

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189

Sa Tm

(a)

(b)

Fig. 9. (a) Three stages of the particle mixing process; (b) particle mixing performance of case 3. Note that the smaller value of mixing index indicates the better mixing quality.

It can be seen that C depends on operation parameters like rotational speed and filling degree. 3.4. Effect of active region on particles mixing performance

y'

x' Fig. 10. Typical velocity vector map for case 3.

Substituting Eqs. (23) and (24) into Eq. (21), the ratio of the active region area to the whole bed area can be expressed as: C¼

1

2L

fπR2

0

∫ δðxÞdx:

ð25Þ

Industrial-scale rotating drum mixers are usually operated in the rolling mode since it provides superior particle mixing. In the rolling mode, particle bed can be briefly divided into two regions, namely the active and static regions. Since the mixing behavior mainly occurs in the active region, particle motion in this region is of prime importance for the mixing performance of the whole drum. The relationship between active region and particles mixing performance is acquired indirectly by investigating the effect of operation parameters such as rotational speed and filling degree on active region and particles mixing performance. The mixing processes of equal-sized ABS beads in a rotating drum at different rotational speeds and filling degrees were simulated by DEM method. The simulation conditions are listed in Table 3. In this study, the rotational speed is varied from 3 rpm to 6 rpm and the filling degree is varied from 20% to 45%. The effects of rotational speed and filling degree on mixing performance are shown in Figs. 13 and 14, respectively. The particle size is 4 mm, particle density is 1100 kg/m3 and the number of these two kinds of particles is equal in simulations. It can be seen that the rotational speed and filling degree have significant influences on the mixing time Tm. The mixing time decreases with the increase of the rotational speed and decrease of the filling degree. The particles reach the uniform state when Tm is about equal

Fig. 11. Simulation results of (a) x-direction velocity profile and (b) y-direction velocity profile for case 3. Note that ω = 0.5236 rad/s, h = 0.2319 m and L = 0.1452 m in case 3.

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Fig. 12. Bed surface velocity profile for case 3.

to 127.2 s in case 1, 100.8 s in case 2, 93.7 s in case 3, 79.3 s in case 4, 43.0 s in case 5, 56.4 s in case 6 and 113.6 s in case 7. As the rotational speed decreases or filling degree increases furthermore, particles are difficultly to reach the uniform state and the mixing time Tm increases multiply. There is no marked change in mixing degree Sa at different rotational speeds and filling degrees. It indicates that the equal-sized particles can reach the uniform state when particles are mixed for a sufficient long time in rotating drum which is operated in the rolling mode. So, industrial-scale rotating drum mixers are usually operated

in the rolling mode. But, there is slight decline of Sa when the filling degree varied from 20% to 45%. This may be due to the fact that at lower filling degree, particles move more freely, and the convective circulation movement is significantly, and particles reach the uniform state easily. However, at the same time, the separation is also obviously, and the mixing index S fluctuated obviously. According to the mixing degree Sa in this study, high filling degree is performed when it is less than 0.5. It also has been observed by Kwapinska et al. [49], who has simulated the mixing process in horizontal rotating drums by DEM in two dimensions at different filling degree. High filling degree means that particles take long time to reach the uniform state. So, reasonable choice of filling degree is very important to enhance particles mixing performance. Fig. 15 presents the thickness of active region δ at different rotational speeds and filling degrees, respectively. We can see that δ is approximately symmetrical with respect to the mid-chord position and the maximum δ appears in x = L which leans in case 5 and 6 may be caused by polynomial fitting error of bed surface velocity u0. Fig. 15(a) shows the maximum δ increases with the rotational speed increasing. As the rotational speed increases, x-direction velocity of particles increase, more particles move into the active region and the active region expands. As the rotational speed further increasing, surface jumping is found, the surface of the active region is not consecutive and particles from the bed being flung into the gas space. Fig. 15(b) shows that the maximum δ decreases with the filling degree increasing. At lower filling degree, the number of particles is smaller, constraints between particles are relatively less and more particles move freely. At higher filling degree, there is no relative motion between most particles. Particles are compacted and collectively rotate as a solid body in static region than motion along the rotating drum wall. However, as the filling degree increases from 20% to

Fig. 13. The mixing performance of particles at different rotational speeds. Note that the smaller value of mixing index indicates the better mixing quality.

X. Xiao et al. / Powder Technology 314 (2017) 182–194

191

Fig. 14. The mixing performance of particles at different filling degrees. Note that the smaller value of mixing index indicates the better mixing quality.

45%, the chord length of particle bed increases which is about equal to 0.256 m in case 5, 0.279 m in case 6, 0.290 m in case 3 and 0.293 in case 7, it is insufficient to describe the size of active region only using the thickness of it when the chord length of particle bed is mutative. The area of active region can be calculated when the thickness of the active region δ is obtained. According to the discussion in Section 3.3.2, the active region area Aa, whole bed area Ab and area ratio C at different rotational speeds and filling degrees are obtained which are presented

in Table 4. We can see that the active region area Aa increases with the increase of rotational speed and the whole bed area Ab maintains a constant value. In addition, the whole bed area Ab increases with the filling degree increasing and the diversification of the active region area Aa can be negligible. Larger whole bed area Ab means more particles are compacted and collectively rotate as a solid body in static region than motion along the rotating drum wall when the active region area Aa is equal. It means that only using the active region area Aa to describe the active region is insufficient, the whole bed area Ab need to be

Fig. 15. Profiles of the thickness of active region δ at (a) different rotational speeds; (b) different filling degrees.

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X. Xiao et al. / Powder Technology 314 (2017) 182–194

Table 4 The results of active region and particles mixing performance at different operation parameters. Case no.

Rotational speed (rpm)

Filling degree

Aa (m2)

Ab (m2)

C

Tm(s)

Sa

1 2 3 4 5 6 7

3 4 5 6 5 5 5

40% 40% 40% 40% 20% 30% 45%

0.0030 0.0048 0.0057 0.0067 0.0068 0.0067 0.0051

0.0272 0.0272 0.0272 0.0272 0.0136 0.0204 0.0305

0.1103 0.1765 0.2096 0.2500 0.5000 0.3284 0.1672

127.2241 100.8337 93.7485 79.3017 43.0321 56.3720 113.6025

0.1398 0.1449 0.1339 0.1316 0.1646 0.1475 0.1305

considered. Thus, the parameter C which associated with Aa and Ab is proposed to characterize the active region. Furthermore, we can conclude that the area ratio C increases with the rotational speed increasing and the filling degree decreasing. In order to investigate the effect of active region on particles mixing performance, the mixing time and mixing degree of particles at different operating parameters also are given out in Table 4. It shows that the change of mixing degree Sa is not obvious which compares with the change of the area ratio C. The results indicate that the area ratio C has significant influence on the mixing time Tm. Figs. 16 and 17 are plotted to describe the relationship between the area ratio and mixing time. It can be seen that the area ratio C increases but the mixing time Tm decreases with the increase of the rotational speed and the decrease of the filling degree. There are marked negative correlation between the area ratio C and the mixing time Tm. It means that particles are more easily to reach the uniform state, mixing time decreases with the increase of the ratio of the active region area to the whole bed area. The higher value of the area ratio C is, the larger the relative size of active region is. More particles move in the active region relatively, and the convective circulation movement is more significantly. So, particles are more likely to reach the uniform mixing state 4. Conclusions The mixing processes of equal-sized ABS beads in a rotating drum were simulated by DEM. A series of experiments such as high rebound test, friction and wear test, stacking angle test and L-box test were designed to measure the contact model parameters including the coefficient of restitution, static friction and rolling friction. Validity of the

Fig. 17. The effect of the area ratio on the mixing time.

DEM simulations was confirmed by comparing with the experimental results. A new parameter C was proposed and defined as the ratio of the active region area to the whole bed area to describe the active region. The effect of the active region on particle mixing performance in rotating drum at rolling mode was investigated. The main conclusions can be summarized as follows: 1. Both of the mixing time Tm and the area ratio C are the function of the operating parameters. The area ratio C increases but the mixing time Tm decreases with the increase of the rotational speed and the decrease of the filling degree. As filling degree increases from 20% to 45%, the maximum thickness of the active region δ decreases and the whole bed area Ab increases. The filling degree has little influences on the active region area Aa. 2. There is no significant change in mixing degree Sa at different rotation speeds and filling degrees. It means that the equal-sized particles can reach the uniform state when particles are mixed for a sufficient long time in rotating drum which is operated in the rolling mode. 3. The area ratio C has significant influence on the mixing time Tm and there is a marked negative correlation between them. The particles are more easily to reach the uniform state with larger

Fig. 16. Plots of the area ratio and mixing time at (a) different rotational speeds; (b) different filling degrees.

X. Xiao et al. / Powder Technology 314 (2017) 182–194

area ratio of the active region to the whole bed. Quantitative relationship between the active region and the particle mixing performance should be investigated furthermore, which is the scope of our future work.

Nomenclature Aa active region area (m2) Ab whole bed area of drum (m2) Ad cross section area of drum (m2) C area ratio e collision coefficient of restitution f ij filling degree ! Fi acting force of particle j to particle i(N) ! Fe external force (N) Fr Froude number g acceleration due to gravity(m s−2) h normal distance between the bed surface center and the drum center of rotation (m) h0 height of rebounding of particle (mm) hs height of particles stacking (mm) H initial height of particle (mm) Ji i rotational inertia of the particle i (kg m2) ! Ke external torque of the particle i (N m) L half-chord length of particle bed (m) mi mass of the particle i (kg) Ni number of particle contacting with the particle i !ij q arm of force from the acting force to the center of the particle i(m) r diameter of platform (mm) ! ri position vector of the particle i (m) R radius of rotating drum (m) S mixing index Sa mixing degree Tm mixing time (s) u x-direction velocity (m s−1) u mean velocity of tracer particle in x direction (m s−1) u0 bed surface velocity (m s−1) ustatic x-direction velocity of particle in static region (m s−1) v y-direction velocity (m s−1) v mean velocity of tracer particle in y direction (m s−1) v1, v2 velocity of two objects after collision (m s−1) v10,v20 velocity of two objects before collision (m s−1) vstatic y-direction velocity of particle in static region (m s−1) x streamwise direction (m) x' horizontal direction (m) y transverse direction (m) y' vertical direction (m) α(x) distance between line of zero velocity and bed surface (m) δ(x) thickness of the active region (m) ε half bed angle of the circular segment occupied with particles (°) θ dynamic angle of repose (°) ! θi angle vector of the particle i μspp static friction coefficient between particles μspw,c critical particle-wall friction coefficient Φ ideal concentration of red particles Φi concentration of red particles in cell i ω rotational speed of drum (rpm) Acknowledgements The authors would like to acknowledge the National Natural Science Foundation of China (No. 51606040 and 51404209), Jiangsu Province Science Foundation for Youths (No. BK20160677), China Postdoctoral

193

Science Foundation (No. 2016M590397) and Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1501001A). Appendix A In Hertz–Mindlin no-slip model, the normal contact forces between particle a and particle b can be described as the function of normal overlap δn: F cn ¼

4  pffiffiffiffiffi 3=2 E R δn 3

ðA:1Þ

where the equivalent Young's Modulus E∗ and the equivalent radius R∗ are defined as: "   #−1 1‐γ 2a 1−γ2b E ¼ þ Ea Eb 

R ¼



1 1 þ Ra Rb

−1

ðA:2Þ

ðA:3Þ

with Ea, γa, Ra and Eb, γb, Rb being the Young's Modulus, Poisson ratio and radius of contact element a and b. The normal damping forces is described as: rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffi rel β Sn m vn 6

F dn ¼ −2

ðA:4Þ

where the equivalent mass m∗, the damping ratio β and the normal stiffness Sn are given by m ¼



1 1 þ ma mb

−1

ðA:5Þ

Ine β ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In2 e þ π 2

ðA:6Þ

pffiffiffiffiffiffiffiffiffiffi R δn

ðA:7Þ

Sn ¼ 2E

with ma and mb being the mass of contact element a and b. e is the coefficient of restitution. vrel n is the normal relative velocity between contact element a and b. The tangential contact forces between particle a and particle b depends on the tangential overlap δt and the tangential stiffness St: F ct ¼ −St δt

ðA:8Þ

pffiffiffiffiffiffiffiffiffiffi St ¼ 8G R δn

ðA:9Þ

where the equivalent shear modulus G∗ is defined as " 

G ¼

  #−1 1‐γ 2a 1−γ2b þ Ga Gb

ðA:10Þ

with Ga and Gb being the shear modulus of contact element a and b. The tangential damping force is described as: rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffi rel β St m vt 6

F dt ¼ −2

ðA:11Þ

is the tangential relative velocities between contact element where vrel t a and b. The tangential force is limited by Coulomb friction according to Fct b μsFcn.

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Rolling friction is accounted for by applying a torque to the contacting surfaces i

T ¼

−μ r F cn Ri ωi

[23]

ðA:12Þ

[24]

where μs and μr are the coefficient of static and rolling friction, respectively.

[25]

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