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Numerical study of the mixing efficiency of a ribbon mixer using the discrete element method
Powder Technology 287 (2016) 380–394
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Numerical study of the mixing efficiency of a ribbon mixer using the discrete element method Gytis Basinskas a,⁎, Mikio Sakai b,⁎ a b
Department of Nuclear Engineering and Management, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Resilience Engineering Research Center, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
a r t i c l e
i n f o
Article history: Received 19 June 2015 Received in revised form 5 October 2015 Accepted 10 October 2015 Available online 22 October 2015 Keywords: Discrete element method Signed distance function Mixing index Ribbon mixer
a b s t r a c t We describe a simulation of granular flow in an industrial ribbon mixer. Such simulations are important to analyze mixing efficiency but they have not been conducted previously in a ribbon mixer, owing to complex boundary conditions. The simulations in this study use the discrete element method (DEM) with a wall boundary model based on a signed distance function (SDF). Introduction of the SDF allows for an accurate representation of the boundary, which is important for proper analysis of granular flow in the ribbon mixer. The adequacy of the DEM/SDF approach was validated qualitatively and quantitatively. The degree of mixing in the ribbon mixer was evaluated using a binary mixture mixing index. The effects of the amount of powder, blade speed and initial loading on the degree of mixing were investigated numerically. In the ribbon mixer, the change of mixing state in the axial direction was found to be much smaller than in the perpendicular direction and an increase in the amount of powder and the blade speed generate better mixing. Only the initial loading and amount of powder were found to be critical for good mixing in the ribbon mixer. The mixing mechanism was investigated by DEM/SDF and an optimal initial loading and amount of powder were suggested for better solid mixing. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The mixing of powders is a critical step to ensure the quality and performance of various products in food, pharmaceutical and chemical engineering industries. A wide variety of industrial mixers exist for different mixture types [1]. Mixing times depend strongly on mixer geometry [2,3] and mixing dynamics can be evaluated experimentally for specific cases [4]. Many experimental approaches can be used to analyze mixing, e.g., sampling, visual tracking, particle image velocimetry (PIV), positron emission particle tracking, magnetic resonance imaging and other techniques [5,6]. However, such approaches may be inaccurate or expensive. The application of numerical simulations is desirable when investigating mixing because they allow for better control of physical properties and a faster analysis. Numerical simulations of particle behavior are often performed by the discrete element method (DEM), which was first derived by Cundall and Strack [7]. Over the years, the DEM method has been improved, modified and coupled with other methods to suit various applications and approaches [8,9]. Solid–fluid interactions can be simulated by using the DEM with smoothed particle hydrodynamics [10], the DEM with a moving particle semi-implicit method [11–13], ⁎ Corresponding author. E-mail addresses: [email protected] (G. Basinskas), [email protected] (M. Sakai).
http://dx.doi.org/10.1016/j.powtec.2015.10.017 0032-5910/© 2015 Elsevier B.V. All rights reserved.
DEM–computational fluid dynamics [14,15] or direct numerical simulation coupled with the DEM [16]. Effects such as agglomeration, breakage and inter-particle bonding can be investigated by using the DEM with a more general contact model [17]. Industrial largescale systems can be simulated using massively parallel DEM approaches [18] or by introducing a coarse-grain model to simplify the system [19–21]. The DEM is also applied to mixing analysis because of its power and flexibility, for example, particle mixing induced by a flat blade [22]. Through the analysis of particle positions, binary mixing can be observed and tracked [23]. The effect of particle size [24] or liquid bonding between particles [25] on powder mixing performance can also be explored using the DEM. Recently, numerical mixing index calculations were performed to analyze solid particle mixing in a plowshare mixer [26], slant cone mixer [27,28] and industrial twinscrew kneader [29]. These researches demonstrated the application of advanced techniques to analyze the effects of initial loading, amount of powder and mixing speed on mixing performance. However, such effects have not been well examined in a ribbon mixer. A ribbon mixer which has a simple casing but complex blade is commonly used to blend industrial dry powders. Experiments have demonstrated that mixing in the axial direction is not as good as mixing in the perpendicular direction, and that the predominant mixing mechanism in the ribbon mixer is convection [30]. Because of the blade complexity, three-dimensional simulations of particle behavior have only been performed recently, e.g., mixing in a single helical ribbon agitator [31] or
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Table 1 Simulation parameters and their values. Parameter
Value
Dimension
Particle diameter d Particle density ρ Spring constant k Particle–particle restitution coefficient eP − P Particle–wall restitution coefficient eP − W Particle–particle friction coefficient μP − P Particle–wall friction coefficient μP − W Integration time step Δt
1.5 2500 1000 0.9 0.9 0.3 0.3 1 × 10−5
mm kg/m3 N/m – – – – s
Fig. 1. Springs, dashpots and a friction slider model for the contact force: a) normal force; b) tangential force.
mixer [8]. The mixing mechanism of a complex ribbon mixer has not been resolved. For example, the degree of mixing and effect of initial loading, amount of powder and mixing speed in a ribbon mixer have not been evaluated in previous studies. To clarify the mixing mechanism, dense granular flow in a complex ribbon mixer has been simulated by the DEM using a signed distance function (SDF)-based wall boundary model [32]. In this study, all analyses are carried out in a mono-dispersed system. Simulations of the DEM/ SDF approach were validated using experimental granular flows observed with a high-speed camera and were evaluated using the PIV method. The degree of mixing was evaluated using the mixing index of a binary mixture. In the evaluations, the effects of amount of powder, blade speed and initial powder loading on the degree of mixing in the ribbon mixer were investigated. 2. Simulation method 2.1. DEM The DEM proposed by Cundall and Strack [7] was used. Solid particle dynamics were calculated using Newton's equations of motion. The translation and rotation of solid particle i are expressed as: Particles X !P−P !P−P Walls X !P−W !P−W ! ! mi r i ¼ F N;i j þ F T;i j þ F N;ik þ F T;ik þ FG j
ð1Þ
k
and Particles X !P−P Walls X !P−W ! I i ω i ¼ T ij þ T ik j
ð2Þ
k
respectively, where mi is the mass of particle i, which is defined with a !P−W ! !P−P !P−P !P−W position vector r l ; F N;i j ; F T;i j ; F N;ik and F T;ik are the normal
Table 2 Simulation and experimental cases. Case
Number of particle number (equivalent mass)
Mixing speed
Experimental validation
1–1 1–2 1–3 2–1 2–2 2–3 3–1 3–2 3–3
400,000 (1767 g) 400,000 (1767 g) 400,000 (1767 g) 700,000 (3092 g) 700,000 (3092 g) 700,000 (3092 g) 1,000,000 (4418 g) 1,000,000 (4418 g) 1,000,000 (4418 g)
20 rpm 40 rpm 60 rpm 20 rpm 40 rpm 60 rpm 20 rpm 40 rpm 60 rpm
no no no no no no yes yes yes
! ! F N and tangential F T forces of particle i interacting with particle j (P– ! ! P) or wall k (P–W); F G is the gravitational force; ω i and Ii are the angular !P−W !P−P are the velocity and moment of inertia of particle i and T i j and T ik ! overall torques T of particle i that interact with particle j or wall k. Finally, all contacts that particle i can make are represented by the summations. The contact force between two particles was modeled using springs, dashpots and a friction slider (Fig. 1). The linear contact model was chosen because a fast algorithm was necessary to calculate a large amount of particles in a reasonable time. Based on our previous studies [11,13, 19] such an approach can be used to simulate realistic particle behavior. In the linear contact model, the stiffness of a spring k, the damping coefficient of a dashpot η and the friction coefficient of a friction slider μ ! are the free parameters that are used to describe the normal F N and ! tangential F T forces: ! ! ! F N ¼ −k δ N −η δ N
ð3Þ
Fig. 2. Geometry of an industrial ribbon mixer with two outer and two inner cylindrical blades (250 mm × 135 mm × 130 mm). The mesh boundary representation as it was used in the simulation. Gravity acts in the direction of the negative y-axis.
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and 8 ! ! ! ! > −k δ T −η δ T ; F T ≤μ F N > > < ! ! ! FT ¼ δ T ! ! > μ F N ! ; F T Nμ F N > > : δ T
! where δ is the displacement of surfaces between two particles.
2.2. SDFs
ð4Þ
Several models exist to create a wall boundary in the DEM. A simple boundary can be created by a combination of planes, where the !P−W displacement is calculated using simple calculaparticle–wall δ tions. However, a complex-shaped wall boundary cannot be created practically using this approach. When a more complex boundary is
Fig. 3. Visual comparison of experimental (left elements) and simulated (right elements) solid flows in Cases 3 (Table 2). The views are of the mixing of 1 million particles at blade speeds of a) 20 rpm, b) 40 rpm, and c) 60 rpm. The quantitative comparisons were performed across the black lines.
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created, representative dummy particles or meshes can be used. A representation of the required boundary with stationary particles is easy and rapid. However, the accuracy of the simulation results may be worse because a smooth surface cannot be created using this approach. Thus meshes are usually used for a complex boundary. In a mesh representation, a boundary is divided into a set of three-dimensional triangles. Collisions are detected by analyzing each of these triangles. When the mesh-based wall boundary is introduced, a more complex detection algorithm must be used since particle–edge, particle–face and particle–vertex interactions must be analyzed [32]. More precise boundary representations require more time to analyze particle collisions. Hence, the existing wall boundary model is problematic from a viewpoint of its applicability. The SDFs for arbitrary boundary conditions were developed to solve the mentioned problem [32]. The advantage of the SDF model is that the collision detection algorithm is unaffected by boundary complexity. The ! signed distance ϕð r Þ is a scalar function defined as the distance to the ! nearest surface from a position vector r . Positive values of the SDF indicate that the distance exists outside the boundary whereas negative values indicate that the distance exists inside the boundary. The boundary itself is represented using a zero contour. It has been shown that several SDFs can be used to represent any complex boundary [32]. When the SDFs are used in a wall boundary, the displacement !P−W between the particle and wall δ can be given by the particle radius ! R and position vector r as follows [32]: !P−W ! δ
r
∇ϕ ! r ! ¼ ϕ r −R : ! ∇ϕ r
The mixing index M is expressed as follow: M¼
σ 20 −σ 2 ; σ 20 −σ 2r
ð8Þ
where σ2 represents the variance of the current mixture. σ2 can be calculated by sampling the particle distribution n times: σ2 ¼
n 2 1X yi −y ; n i¼1
ð9Þ
where yi is the fraction of the chosen particle type in sample i and y is the average fraction of the chosen particle in the mixture. These fractions can be defined using either type-A or type-B particles. yi and y for type-A particles can be defined as: yi ¼
NiA N iA
þ NiB
ð10Þ
and y¼
NA NA þ NB
ð11Þ
when, type-B particles are used as a reference, the definitions are: yi ¼
ð5Þ
383
NiB N iA
þ NiB
ð12Þ
! ! The normalized gradient of the SDF ∇ϕð r Þ=j∇ϕð r Þj is used to calculate the normal vector of the boundary that is used to define the dis! placement direction. The SDF ϕð r Þ is set initially at discrete sampling points and during the simulation; other points are interpolated from the nearby values. The SDFs of the boundary can be defined based on a usual mesh representation before the calculation, and recalculation is thus unnecessary during solid flow calculations. 2.3. Mixing index The mixing state of the granular flow in the ribbon mixer was analyzed by calculating the mixing index M. Of the many mixing indices available [6], the Lacey index [33] was chosen because it is the most popular. In addition, this mixing index has a finite range, which is important for an adequate comparison of different mixtures. Lacey's approach is based on a change of mixing variance to the greatest change of mixing variance of a binary mixture; the value of the mixing index ranges from 0 to 1, where 0 represents a segregated state (with variance σ20), whereas 1 represents a perfectly mixed state (with varianceσ2r ). By dividing the particles into two types (for example types A and B) and by taking into account that the particles have the same mass and size it is possible to define σ20 and σ2r variances: σ 20 ¼
NA NB ðNA þ NB Þ2
ð6Þ
and σ 2r ¼
σ 20 Nrand
ð7Þ
where NA and NB are the number of type A and type B particles, respectively, and Nrand is the number of particles in a random sample.
Fig. 4. Average particle speed measured at different positions along the x (a) and z (b) direction (lines at Fig. 3) by averaging one full rotation at each analyzed mixing speed. 1 million particles mixing at 20, 40 and 60 rpm speed (Cases 3).
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Fig. 5. Mixing states at different time (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles at blade speed of 20 rpm. Particles were classified along the x-axis.
and y¼
NB ; NA þ NB
ð13Þ
Adequacy of the calculation results was backed by qualitative and quantitative analysis of the simulated and experimental particle velocity distributions. 3.1. Simulation and experimental conditions
where NiA and NiB are the numbers of type-A and type-B particles in sample i, respectively. Important factors affecting the mixing index are the sample size and number of samples. In addition, the direction of particle classification is important depending on what type of mixing is analyzed. It is possible to use a modified generalized mean mixing index (similar to what was used by Siraj et al. [24]), which does not require a classification direction. However, this study investigates axis-dependent mixing and therefore uses the approach discussed earlier. 3. Validation Validation tests were conducted at different mixing speeds to prove the applicability of the DEM/SDF in the ribbon mixer system.
We used an industrial ribbon mixer (with casing dimensions of 250 mm, 135 mm and 130 mm in the x-, y- and z-directions, respectively) to simulate and validate the granular flow. The ribbon mixer contains two inner (74 mm diameter, 11 mm width) and two outer (114 mm diameter, 7 mm width) helical blades, which are attached to a central rotating shaft by small rods (6 mm diameter) as illustrated in Fig. 2. The blades are ~ 3 mm thick with rounded endings. The mixing speed can be set to higher than 120 rpm but up to half of that value was used in this study. The recommended initial powder fill was ~ 50% of the tank volume, which equates to ~ 3 kg or ~ 700,000 particles. Glass beads (1.5 mm diameter, 2500 kg/m3 density, UB-2223LN, UNION Co. Ltd., Osaka, Japan) were used in the validation tests. The
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Fig. 6. Mixing states at different time (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles at blade speed of 40 rpm. Particles were classified along the x-axis.
physical properties used in the simulations and experiments are shown in Table 1. The spring constant k, coefficient of restitution e and friction μ were set to 1000 N/m, 0.9 and 0.3, respectively. These values were chosen after a sensitivity analysis and comparison with experiments from our previous work [11,13,19]. To our knowledge, these values represent the most realistic particle behavior. The damping coefficient η is defined through the restitution coefficient e [34]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk ; 2 ð ln eÞ −π2
η ¼ 2 ln e
ð14Þ
where m is the particle mass. In this study, the restitution coefficient of 0.9 is equivalent to a damping coefficient of 4.46 × 10−3 Ns/m. The value of the simulation time step Δt is important for accurate integration of the motion equations. For the linear spring
model, Tsuji et al. [34] proposed an inequality as an adequate time step Δt: Δt ≤
π 5
rffiffiffiffiffi m : k
ð15Þ
In this study, the critical value for the time step Δt is ~ 4 × 10−5 s which is four times greater than the value of 1 × 10−5 s that we used. One million particles (4.418 kg total mass) were used in the validation tests as shown in Table 2. The rotational speed of the blade was set to 20, 40 and 60 rpm. Granular motion was recorded using a Phantom v9.1 high-speed camera (Vision Research Inc., Wayne, NJ, USA). Experimental data were analyzed using PIV implemented in the DynamicStudio (Dantec Dynamics, Nova Instruments, Woburn, MA, USA) to evaluate the granular flow of particles. Visual data were segmented into 27 by 52 segments and analyzed. Several photographs
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were taken at different times and the velocity of each segment was evaluated using the PIV method. In the simulation case, a similar procedure was also implemented, except that the segment velocity was evaluated by averaging the nearby particle velocities. All simulations were carried out using multi-thread parallelized DEM on a single personal computer. The computational code was developed in our laboratory [35] and was validated in our previous studies [11,13,19]. The time required for the simulation of one full time step are ~0.15, 0.22 and 0.38 s for 400,000, 700,000 and 1,000,000 particle cases, respectively. 3.2. Results and discussion Three validation tests were performed. Velocity distributions of the solid phase because of different mixing speeds were compared qualitatively and quantitatively between the simulation and experiment. The comparison was made by observing the granular flow at the top of the mixing device. Fig. 3 shows the velocity distribution of the solid phase obtained from the simulation and experimental results in Cases 3–1, 3–2 and 3–3. Convective flows were observed in all cases. The tendency to granular flow, namely, the direction and magnitude of the velocity
distribution, agreed well between the simulation and experimental results. In all cases, solid particles near the blades achieved relatively higher velocities. Granular flow increases in a similar manner to the increase in rotational speed. The highest speeds increase from 0.2 to 0.5 m/s with an increase in blade speed from 20 to 60 rpm (note the different scales in Fig. 3). For a quantitative comparison, the average particle speed was measured across x and z directions (lines at Fig. 3) at several discretized positions by averaging one full rotation at each analyzed mixing speed. The average particle speed was chosen since particle movement is mostly one directional. The simulated and experimental values of the average particle speed are shown in Fig. 4. The average particle speeds agree relatively well between the simulations and experiments in all analyzed mixing speeds. The highest average particle speed exists in the middle of the mixer and increases from ~0.08 m/s to 0.25 m/s with increase in mixing speed from 20 rpm to 60 rpm. The average particle speed decreases (~50% of the central value) similarly (almost identical in the simulations) from the center to both sides along the z direction (Fig. 4b). Along the x direction (Fig. 4a) this decrease is much higher and not symmetrical. This effect is connected to the avalanche of particles along the x direction. The highest average particle speed in the center of the mixer can be explained by considering that
Fig. 7. Mixing states at different time (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles at blade speed of 60 rpm. Particles were classified along the x-axis.
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the two sets of blades intersect at that point (Fig. 2) to give more particle movement in one rotation compared with the sides. In summary, qualitative and quantitative agreement between the computed and experimental results for different mixing speeds implies that the DEM/SDF is adequate to simulate granular flow in the ribbon mixer. Convective granular flow could be simulated using a simple linear spring system, and the complex-shaped mixer surface could be modeled by the SDF. Consequently, adequacy of the DEM/SDF was shown through the validation tests. This indicates that the DEM/SDF can simulate detailed granular flow in the ribbon mixer. Therefore, DEM/SDF is applied to investigate solid mixing efficiency in the ribbon mixer. 4. Mixing mechanism in the ribbon mixer Mixing states can be evaluated based on particle positions in the ribbon mixer. To investigate the mixing efficiency of solid particles in the ribbon mixer, DEM simulations were performed by varying the blade speed and particle number. Central shaft speeds were chosen as 20, 40 and 60 rpm and the number of granules was 400,000, 700,000 and 1,000,000. Therefore particle behavior was simulated for nine cases (see Table 2) and 20 s of physical time. The physical properties were the same as those used in the validation tests (Table 1). Particles were distinguished using different colors, namely, black and white, where
387
the number of each was equivalent. Simulation cases were compared visually using the colored particle distributions. A visual observation of the mixing behavior reveals the scope of mixing and the different axial coloring of the particles helps to clarify the different mixing behaviors. Lacey's mixing index was used to evaluate these mixing states numerically.
4.1. Visual observations of mixing The effect of the number of particles and blade speed on mixing efficiency was investigated. Fig. 5 shows the mixing states at different time steps for an increasing number of particles whereas the mixing speed was 20 rpm in all calculations (Cases 1–1, 2–1 and 3–1). The same cases are shown in Figs. 6 and 7 for mixing speeds of 40 rpm (Cases 1–2, 2–2 and 3–2) and 60 rpm (Cases 1–3, 2–3 and 3–3), respectively. Better mixing can be achieved using more particles and faster blade speeds. For the 400,000 and 700,000 particles, masses of white and black particles existed even after 20 s of mixing for all speeds. When 1,000,000 particles were used, the black and white particles seemed to achieve a homogenous distribution only after 20 s and at 60 rpm speed. Therefore, the mixing efficiency is influenced by the amount of powder and the blade speed.
Fig. 8. Mixing states for Case 4–3 at different times (0, 5, 10 and 20 s from top down) using particle classification along the x-, y- and z-axes.
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Fig. 9. Mixing states at different times (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles with blade speed of 60 rpm using particle classification along the z-axis.
The effects of initial powder loading were analyzed by classifying particles in different directions. In Figs. 5, 6, and 7 the classification is only along the x-axis. In Fig. 8, classifications along the x-, y- and z-
axes are used for 1 million particles mixed at 60 rpm (Case 3–3). After some time, the y-axis mixing state is similar to the x-axis mixing state. However, z-axis mixing is very different and particles remain unmixed even after a long time. Furthermore, the use of different particle numbers does not generate a good mixture along the z-axis. Fig. 9 shows similar states of a low degree of mixing along the z-axis for all analyzed particle numbers (Cases 1–3, 2–3 and 3–3). It is clear that in this mixer, the diffusion effect in the z-direction is quite small. 4.2. Mixing states
Fig. 10. Mixing indices for different sampling volumes in Case 3–3. Particle classification is along the x-axis.
Although a visual comparison gives a good idea of the mixing scope, it is insufficient. The use of a mixing index allows for a quantitative evaluation of the mixing state. The sample volume is important when analyzing the mixing index [1]. Different sample volumes were tested before the calculations were conducted using the mixing index in Case 3–3 (1 million particles mixing at 60 rpm). The sample volume was expressed as a ratio of sample volume to particle volume. Several ratios ranging from 2 to 1644 were tested. The results are shown in Fig. 10, where it is seen that the mixing index increases with sample volume. Given that almost all particles are sampled, larger samples can have
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Fig. 11. Effect of blade speed on the mixing state (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the x-axis.
larger chunks to achieve the same mixing index as smaller samples. Therefore, the mixing index increases. For subsequent calculations, the ratio was chosen to be 247, which corresponds to a cube with sides of approximately 10 mm. Such a ratio was chosen to contain sufficient number of samples with adequate sample size. If the amount of particles in a sample is less than 70% of the closely packed sphere number, then the sample is discarded. The average number of samples is ~ 1500, 2450 and 3200 for the 400,000, 700,000 and 1,000,000 particle cases, respectively, which give a standard error of measured variance of ~3.7%,
Fig. 12. Effect of amount of powder on the mixing state (blade speeds of 20, 40 and 60 rpm for 400,000, 700,000 and 1,000,000 particles). Particle classification is along the x-axis.
2.9% and 2.6%, respectively. Such values show that our calculated mixing indices are sufficiently accurate to be compared across cases. 4.2.1. Effect of blade speed Fig. 11 shows the change in mixing index for all the simulated cases by grouping cases with the same number of particles. The higher blade
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Fig. 13. Effect of blade speed on the mixing state (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the y-axis.
Fig. 14. Effect of blade speed on the mixing state (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the z-axis.
speeds generate better mixing for all the analyzed particle numbers (cases are compared visually in Figs. 5, 6 and 7). Therefore, convective mixing can be enhanced using a higher mixing speed.
4.2.2. Effect of amount of powder The effect of the amount of powder can be analyzed by grouping the mixing indices with the same blade speed as shown in Fig. 12. Better mixing is achieved by increasing the number of particles. Such an effect was also observed by comparing the particle distribution visually
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Fig. 15. Effect of amount of powder on the mixing state (blade speeds of 20, 40 and 60 rpm for 400,000, 700,000 and 1,000,000 particles). Particle classification is along the z-axis.
(Figs. 5, 6 and 7). Better mixing states in 700,000 and 1,000,000 particle cases indicate that without a sufficient amount of powder, good mixing performance cannot be achieved even after a long time.
391
Fig. 16. Mixing index versus number of blade rotations (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the x-axis.
4.2.3. Effect of initial powder location As was done in the visual analysis using different axis classifications, the effect of initial powder loading can be investigated. An analysis of the mixing index shows that mixing in the y-axis direction (Fig. 13) is very similar to mixing in the x-axis direction (Fig. 11). This occurs
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Fig. 18. Average axis-mixing of 400,000, 700,000 and 1,000,000 particles at blade speed of 60 rpm.
of particle distributions along different axes (Figs. 8 and 9). It therefore appears that the mixing of 1,000,000 particles is no longer the best option. This can be seen better by grouping the same blade speed in Fig. 15. The mixing of 1 million particles is slower than that of a smaller number of particles, and this tendency strengthens with increase in blade speed. In addition, the mixing of 700,000 particles is better than that of 400,000 particles. The mixing efficiency with 400,000 particles decreases with an increase in mixing speed compared with that for 700,000 particles. Mixing along the z-axis is extremely low. Thus, an increase in mixing degree along this axis using 700,000 particles may be irrelevant. Such a behavior may also be associated with a different mixing mechanism, namely, shear mixing. Similar results, where an increase in powder reduced the mixing efficiency, were also observed in recent studies of slant cone mixers [27] and industrial twin-screw kneaders [29]. 4.2.4. Mixing per number of blade rotations Another comparison between mixing indices can be achieved by expressing the mixing index versus number of blade rotations and not time. The mixing index versus number of rotations for particles classified along the x-axis and grouping by particle number is shown in Fig. 16. Each group's mixing index is expressed in such a way that they are almost identical. The high mixing rate starts to decline after approximately five blade rotations in all cases. The plateau, which the mixing rate reaches, increases with number of particles. Therefore, only mixing speed reduces the required mixing time. Since particle behavior in the x- and y-directions is similar, similar results can be obtained. However, mixing in the z-direction is different. The mixing index expressed by a number of blade rotations using particle classification along this direction exhibits similar effects (Fig. 17). However, in this case, the mixing index increases linearly in all the analyzed particle numbers. It is unclear when the mixing index reaches a plateau or if the blade speed affects it. Extrapolation of the mixing indices with the lowest analyzed mixing speed to compare it with the highest is unclear since small differences in the inclinations of the mixing lines are within the mixing error. It can be said with certainty, however, that the highest measured mixing indices exist for the 700,000 particle cases. Fig. 17. Mixing index versus number of blade rotations (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the z-axis.
because these two directions are perpendicular to the rotation axis of the blades. However, mixing along the z-axis (which is the same as the rotation axis) is different. Mixing indices along the z-axis are shown in Fig. 14. It can be seen that the mixing index increases almost linearly with time. These results agree well with the visual comparison
4.2.5. Overall mixing In summary, mixing along the axes perpendicular to the rotation axis is better and increases with increase in speed and number of particles. Mixing in the axial direction is extremely slow, and increases for the 700,000-particle case. The overall mixing behavior can be represented by plotting the average axis-mixing index at 60 rpm with different numbers of particles (Cases 1–3, 2–3 and 3–3) (Fig. 18). The overall mixing of 700,000 particles improved over the 1,000,000 particle mixing after ~ 15 s, whereas the 400,000-particle case produced the
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lowest mixing performance. The main reason why the mixing of 700,000 particles was suggested as the optimal case can be explained by considering the significant increase in mixing of the 700,000 particles in the axial direction compared with that of the 1,000,000 particle mixing. It can be argued that an increase in mixing performance in the axis perpendicular to the axial direction can be associated with convective mixing, whereas a sudden decrease in mixing performance for the 1,000,000 particle in the axial direction shows shearing mixing effects. This may occur because of two separate blades in this direction (see Fig. 2). These blades are inverted, which results in an ineffective convection mixing and an increased effect of shear mixing. It is clear that in such a ribbon mixer, the design of the powder loading is important. 5. Conclusions In this study, the DEM with a wall boundary model based on SDF was used to simulate granular flow in a ribbon mixer. The simulations were validated qualitatively and quantitatively using experiments. The effects of amount of powder, blade speed and initial loading on the degree of mixing were investigated. The following conclusions can be made: a) This study demonstrates that granular flow in a ribbon mixer can be simulated by the DEM/SDF. b) Adequacy of the DEM/SDF was demonstrated through a qualitative and quantitative comparison between the simulated and experimental results. The simulated and experimental velocity distributions and average particle speeds agreed well for all the analyzed mixing speeds. c) An analysis of the simulation data shows that, for the analyzed ribbon mixer, the amount of powder and the blade speed are important factors that affect the mixing quality. The mixture quality increases with an increase in the amount of powder and the blade speed. However, the amount of powder is more important since, without a sufficient amount, a good mixture cannot be achieved even at high mixing speeds. d) The mixing index expressed per number of blade rotations does not depend on the mixing speed. e) An analysis of axis-dependent mixing reveals the importance of the initial location of the powder since particle effects depend on the axis orientation. The mixing rate in the axial direction is extremely low compared with that in the perpendicular directions. f) In the axial direction, optimal mixing can be achieved with 700,000 particles, whereas in the perpendicular directions, such mixing can only be achieved using 1,000,000 particles. Such an effect explains the improved average axis-mixing performance of 700,000 particles. g) A visual evaluation of the mixing behavior agrees well with the mixing index evaluation. This work shows that DEM techniques are starting to reach application at industrial level. A mixing analysis using the DEM/SDF approach can be applied easily to other mixers to provide insightful results.
Nomenclature Roman letters d particle diameter (m) e coefficient of restitution (−) ! F contact force (N) ! FG gravitational force (N) I moment of inertia of a particle (kg m2) k spring constant (N/m) M mixing index (−) m particle mass (kg) N number of particles (−) n sample size (−) R particle radius (m) ! r ðtÞ position vector of a particle (m)
! T t Δt y y
393
torque (N m) time (s) time step of the simulation (s) fraction of chosen particles in a sample (−) average fraction of chosen particles in all samples (−)
Greek letters ! δ displacement (m) η coefficient of damping (Ns/m) μ coefficient of friction (−) ρ particle density (kg/m3) σ standard deviation (−) φ signed distance function (m) ! ω particle angular velocity (rad/s) Subscripts i particle index j index of a particle interacting with particle i k wall index N normal component T tangential component 0 fully segregated state r fully mixed state rand random sample A type-A particles B type-B particles Superscripts i sample index P–P particle–particle interaction P–W particle–wall interaction Operator ∇ gradient operator (m−1) Acknowledgments Part of this study was supported financially by MAKINO MFG. Co., Ltd. The authors are grateful to Dr. Yusuke Shigeto for his kind support. References [1] J. Bridgwater, Mixing of powders and granular materials by mechanical means—a perspective, Particuology 10 (2012) 397–427, http://dx.doi.org/10.1016/j.partic. 2012.06.002. [2] S. Masiuk, Power consumption, mixing time and attrition action for solid mixing in a ribbon mixer, Powder Technol. 51 (1987) 217–229, http://dx.doi.org/10.1016/00325910(87)80022-0. [3] S. Masiuk, H. Łacki, F. Strek, Power consumption and mixing times for liquid mixing in a ribbon mixer, Chem. Eng. J. 48 (1992) 135–140, http://dx.doi.org/10.1016/ 0300-9467(92)85015-2. [4] J. Thyn, K. Duffek, Powder mixing in a horizontal batch mixer, Powder Technol. 15 (1975) 193–197, http://dx.doi.org/10.1016/0032-5910(76)80048-4. [5] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulate systems: a review of major applications and findings, Chem. Eng. Sci. 63 (2008) 5728–5770, http://dx.doi.org/10.1016/j.ces.2008.08.006. [6] A.N. Huang, H.P. Kuo, Developments in the tools for the investigation of mixing in particulate systems — a review, Adv. Powder Technol. 25 (2014) 163–173, http:// dx.doi.org/10.1016/j.apt.2013.10.007. [7] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1979) 47–65. [8] F. Bertrand, L.A. Leclaire, G. Levecque, DEM-based models for the mixing of granular materials, Chem. Eng. Sci. 60 (2005) 2517–2531, http://dx.doi.org/10.1016/j.ces. 2004.11.048. [9] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulate systems: theoretical developments, Chem. Eng. Sci. 62 (2007) 3378–3396, http://dx. doi.org/10.1016/j.ces.2006.12.089. [10] X. Sun, M. Sakai, Y. Yamada, Three-dimensional simulation of a solid–liquid flow by the DEM–SPH method, J. Comput. Phys. 248 (2013) 147–176, http://dx.doi.org/10. 1016/j.jcp.2013.04.019. [11] M. Sakai, Y. Shigeto, J. Xiong, T. Saito, X. Sun, S. Koshizuka, et al., Lagrangian– Lagrangian modeling for a solid–liquid flow in a cylindrical tank, Chem. Eng. J. 200–202 (2012) 663–672, http://dx.doi.org/10.1016/j.cej.2012.06.080.
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Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
Numerical study of the mixing efficiency of a ribbon mixer using the discrete element method Gytis Basinskas a,⁎, Mikio Sakai b,⁎ a b
Department of Nuclear Engineering and Management, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Resilience Engineering Research Center, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
a r t i c l e
i n f o
Article history: Received 19 June 2015 Received in revised form 5 October 2015 Accepted 10 October 2015 Available online 22 October 2015 Keywords: Discrete element method Signed distance function Mixing index Ribbon mixer
a b s t r a c t We describe a simulation of granular flow in an industrial ribbon mixer. Such simulations are important to analyze mixing efficiency but they have not been conducted previously in a ribbon mixer, owing to complex boundary conditions. The simulations in this study use the discrete element method (DEM) with a wall boundary model based on a signed distance function (SDF). Introduction of the SDF allows for an accurate representation of the boundary, which is important for proper analysis of granular flow in the ribbon mixer. The adequacy of the DEM/SDF approach was validated qualitatively and quantitatively. The degree of mixing in the ribbon mixer was evaluated using a binary mixture mixing index. The effects of the amount of powder, blade speed and initial loading on the degree of mixing were investigated numerically. In the ribbon mixer, the change of mixing state in the axial direction was found to be much smaller than in the perpendicular direction and an increase in the amount of powder and the blade speed generate better mixing. Only the initial loading and amount of powder were found to be critical for good mixing in the ribbon mixer. The mixing mechanism was investigated by DEM/SDF and an optimal initial loading and amount of powder were suggested for better solid mixing. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The mixing of powders is a critical step to ensure the quality and performance of various products in food, pharmaceutical and chemical engineering industries. A wide variety of industrial mixers exist for different mixture types [1]. Mixing times depend strongly on mixer geometry [2,3] and mixing dynamics can be evaluated experimentally for specific cases [4]. Many experimental approaches can be used to analyze mixing, e.g., sampling, visual tracking, particle image velocimetry (PIV), positron emission particle tracking, magnetic resonance imaging and other techniques [5,6]. However, such approaches may be inaccurate or expensive. The application of numerical simulations is desirable when investigating mixing because they allow for better control of physical properties and a faster analysis. Numerical simulations of particle behavior are often performed by the discrete element method (DEM), which was first derived by Cundall and Strack [7]. Over the years, the DEM method has been improved, modified and coupled with other methods to suit various applications and approaches [8,9]. Solid–fluid interactions can be simulated by using the DEM with smoothed particle hydrodynamics [10], the DEM with a moving particle semi-implicit method [11–13], ⁎ Corresponding author. E-mail addresses: [email protected] (G. Basinskas), [email protected] (M. Sakai).
http://dx.doi.org/10.1016/j.powtec.2015.10.017 0032-5910/© 2015 Elsevier B.V. All rights reserved.
DEM–computational fluid dynamics [14,15] or direct numerical simulation coupled with the DEM [16]. Effects such as agglomeration, breakage and inter-particle bonding can be investigated by using the DEM with a more general contact model [17]. Industrial largescale systems can be simulated using massively parallel DEM approaches [18] or by introducing a coarse-grain model to simplify the system [19–21]. The DEM is also applied to mixing analysis because of its power and flexibility, for example, particle mixing induced by a flat blade [22]. Through the analysis of particle positions, binary mixing can be observed and tracked [23]. The effect of particle size [24] or liquid bonding between particles [25] on powder mixing performance can also be explored using the DEM. Recently, numerical mixing index calculations were performed to analyze solid particle mixing in a plowshare mixer [26], slant cone mixer [27,28] and industrial twinscrew kneader [29]. These researches demonstrated the application of advanced techniques to analyze the effects of initial loading, amount of powder and mixing speed on mixing performance. However, such effects have not been well examined in a ribbon mixer. A ribbon mixer which has a simple casing but complex blade is commonly used to blend industrial dry powders. Experiments have demonstrated that mixing in the axial direction is not as good as mixing in the perpendicular direction, and that the predominant mixing mechanism in the ribbon mixer is convection [30]. Because of the blade complexity, three-dimensional simulations of particle behavior have only been performed recently, e.g., mixing in a single helical ribbon agitator [31] or
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Table 1 Simulation parameters and their values. Parameter
Value
Dimension
Particle diameter d Particle density ρ Spring constant k Particle–particle restitution coefficient eP − P Particle–wall restitution coefficient eP − W Particle–particle friction coefficient μP − P Particle–wall friction coefficient μP − W Integration time step Δt
1.5 2500 1000 0.9 0.9 0.3 0.3 1 × 10−5
mm kg/m3 N/m – – – – s
Fig. 1. Springs, dashpots and a friction slider model for the contact force: a) normal force; b) tangential force.
mixer [8]. The mixing mechanism of a complex ribbon mixer has not been resolved. For example, the degree of mixing and effect of initial loading, amount of powder and mixing speed in a ribbon mixer have not been evaluated in previous studies. To clarify the mixing mechanism, dense granular flow in a complex ribbon mixer has been simulated by the DEM using a signed distance function (SDF)-based wall boundary model [32]. In this study, all analyses are carried out in a mono-dispersed system. Simulations of the DEM/ SDF approach were validated using experimental granular flows observed with a high-speed camera and were evaluated using the PIV method. The degree of mixing was evaluated using the mixing index of a binary mixture. In the evaluations, the effects of amount of powder, blade speed and initial powder loading on the degree of mixing in the ribbon mixer were investigated. 2. Simulation method 2.1. DEM The DEM proposed by Cundall and Strack [7] was used. Solid particle dynamics were calculated using Newton's equations of motion. The translation and rotation of solid particle i are expressed as: Particles X !P−P !P−P Walls X !P−W !P−W ! ! mi r i ¼ F N;i j þ F T;i j þ F N;ik þ F T;ik þ FG j
ð1Þ
k
and Particles X !P−P Walls X !P−W ! I i ω i ¼ T ij þ T ik j
ð2Þ
k
respectively, where mi is the mass of particle i, which is defined with a !P−W ! !P−P !P−P !P−W position vector r l ; F N;i j ; F T;i j ; F N;ik and F T;ik are the normal
Table 2 Simulation and experimental cases. Case
Number of particle number (equivalent mass)
Mixing speed
Experimental validation
1–1 1–2 1–3 2–1 2–2 2–3 3–1 3–2 3–3
400,000 (1767 g) 400,000 (1767 g) 400,000 (1767 g) 700,000 (3092 g) 700,000 (3092 g) 700,000 (3092 g) 1,000,000 (4418 g) 1,000,000 (4418 g) 1,000,000 (4418 g)
20 rpm 40 rpm 60 rpm 20 rpm 40 rpm 60 rpm 20 rpm 40 rpm 60 rpm
no no no no no no yes yes yes
! ! F N and tangential F T forces of particle i interacting with particle j (P– ! ! P) or wall k (P–W); F G is the gravitational force; ω i and Ii are the angular !P−W !P−P are the velocity and moment of inertia of particle i and T i j and T ik ! overall torques T of particle i that interact with particle j or wall k. Finally, all contacts that particle i can make are represented by the summations. The contact force between two particles was modeled using springs, dashpots and a friction slider (Fig. 1). The linear contact model was chosen because a fast algorithm was necessary to calculate a large amount of particles in a reasonable time. Based on our previous studies [11,13, 19] such an approach can be used to simulate realistic particle behavior. In the linear contact model, the stiffness of a spring k, the damping coefficient of a dashpot η and the friction coefficient of a friction slider μ ! are the free parameters that are used to describe the normal F N and ! tangential F T forces: ! ! ! F N ¼ −k δ N −η δ N
ð3Þ
Fig. 2. Geometry of an industrial ribbon mixer with two outer and two inner cylindrical blades (250 mm × 135 mm × 130 mm). The mesh boundary representation as it was used in the simulation. Gravity acts in the direction of the negative y-axis.
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and 8 ! ! ! ! > −k δ T −η δ T ; F T ≤μ F N > > < ! ! ! FT ¼ δ T ! ! > μ F N ! ; F T Nμ F N > > : δ T
! where δ is the displacement of surfaces between two particles.
2.2. SDFs
ð4Þ
Several models exist to create a wall boundary in the DEM. A simple boundary can be created by a combination of planes, where the !P−W displacement is calculated using simple calculaparticle–wall δ tions. However, a complex-shaped wall boundary cannot be created practically using this approach. When a more complex boundary is
Fig. 3. Visual comparison of experimental (left elements) and simulated (right elements) solid flows in Cases 3 (Table 2). The views are of the mixing of 1 million particles at blade speeds of a) 20 rpm, b) 40 rpm, and c) 60 rpm. The quantitative comparisons were performed across the black lines.
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created, representative dummy particles or meshes can be used. A representation of the required boundary with stationary particles is easy and rapid. However, the accuracy of the simulation results may be worse because a smooth surface cannot be created using this approach. Thus meshes are usually used for a complex boundary. In a mesh representation, a boundary is divided into a set of three-dimensional triangles. Collisions are detected by analyzing each of these triangles. When the mesh-based wall boundary is introduced, a more complex detection algorithm must be used since particle–edge, particle–face and particle–vertex interactions must be analyzed [32]. More precise boundary representations require more time to analyze particle collisions. Hence, the existing wall boundary model is problematic from a viewpoint of its applicability. The SDFs for arbitrary boundary conditions were developed to solve the mentioned problem [32]. The advantage of the SDF model is that the collision detection algorithm is unaffected by boundary complexity. The ! signed distance ϕð r Þ is a scalar function defined as the distance to the ! nearest surface from a position vector r . Positive values of the SDF indicate that the distance exists outside the boundary whereas negative values indicate that the distance exists inside the boundary. The boundary itself is represented using a zero contour. It has been shown that several SDFs can be used to represent any complex boundary [32]. When the SDFs are used in a wall boundary, the displacement !P−W between the particle and wall δ can be given by the particle radius ! R and position vector r as follows [32]: !P−W ! δ
r
∇ϕ ! r ! ¼ ϕ r −R : ! ∇ϕ r
The mixing index M is expressed as follow: M¼
σ 20 −σ 2 ; σ 20 −σ 2r
ð8Þ
where σ2 represents the variance of the current mixture. σ2 can be calculated by sampling the particle distribution n times: σ2 ¼
n 2 1X yi −y ; n i¼1
ð9Þ
where yi is the fraction of the chosen particle type in sample i and y is the average fraction of the chosen particle in the mixture. These fractions can be defined using either type-A or type-B particles. yi and y for type-A particles can be defined as: yi ¼
NiA N iA
þ NiB
ð10Þ
and y¼
NA NA þ NB
ð11Þ
when, type-B particles are used as a reference, the definitions are: yi ¼
ð5Þ
383
NiB N iA
þ NiB
ð12Þ
! ! The normalized gradient of the SDF ∇ϕð r Þ=j∇ϕð r Þj is used to calculate the normal vector of the boundary that is used to define the dis! placement direction. The SDF ϕð r Þ is set initially at discrete sampling points and during the simulation; other points are interpolated from the nearby values. The SDFs of the boundary can be defined based on a usual mesh representation before the calculation, and recalculation is thus unnecessary during solid flow calculations. 2.3. Mixing index The mixing state of the granular flow in the ribbon mixer was analyzed by calculating the mixing index M. Of the many mixing indices available [6], the Lacey index [33] was chosen because it is the most popular. In addition, this mixing index has a finite range, which is important for an adequate comparison of different mixtures. Lacey's approach is based on a change of mixing variance to the greatest change of mixing variance of a binary mixture; the value of the mixing index ranges from 0 to 1, where 0 represents a segregated state (with variance σ20), whereas 1 represents a perfectly mixed state (with varianceσ2r ). By dividing the particles into two types (for example types A and B) and by taking into account that the particles have the same mass and size it is possible to define σ20 and σ2r variances: σ 20 ¼
NA NB ðNA þ NB Þ2
ð6Þ
and σ 2r ¼
σ 20 Nrand
ð7Þ
where NA and NB are the number of type A and type B particles, respectively, and Nrand is the number of particles in a random sample.
Fig. 4. Average particle speed measured at different positions along the x (a) and z (b) direction (lines at Fig. 3) by averaging one full rotation at each analyzed mixing speed. 1 million particles mixing at 20, 40 and 60 rpm speed (Cases 3).
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Fig. 5. Mixing states at different time (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles at blade speed of 20 rpm. Particles were classified along the x-axis.
and y¼
NB ; NA þ NB
ð13Þ
Adequacy of the calculation results was backed by qualitative and quantitative analysis of the simulated and experimental particle velocity distributions. 3.1. Simulation and experimental conditions
where NiA and NiB are the numbers of type-A and type-B particles in sample i, respectively. Important factors affecting the mixing index are the sample size and number of samples. In addition, the direction of particle classification is important depending on what type of mixing is analyzed. It is possible to use a modified generalized mean mixing index (similar to what was used by Siraj et al. [24]), which does not require a classification direction. However, this study investigates axis-dependent mixing and therefore uses the approach discussed earlier. 3. Validation Validation tests were conducted at different mixing speeds to prove the applicability of the DEM/SDF in the ribbon mixer system.
We used an industrial ribbon mixer (with casing dimensions of 250 mm, 135 mm and 130 mm in the x-, y- and z-directions, respectively) to simulate and validate the granular flow. The ribbon mixer contains two inner (74 mm diameter, 11 mm width) and two outer (114 mm diameter, 7 mm width) helical blades, which are attached to a central rotating shaft by small rods (6 mm diameter) as illustrated in Fig. 2. The blades are ~ 3 mm thick with rounded endings. The mixing speed can be set to higher than 120 rpm but up to half of that value was used in this study. The recommended initial powder fill was ~ 50% of the tank volume, which equates to ~ 3 kg or ~ 700,000 particles. Glass beads (1.5 mm diameter, 2500 kg/m3 density, UB-2223LN, UNION Co. Ltd., Osaka, Japan) were used in the validation tests. The
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Fig. 6. Mixing states at different time (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles at blade speed of 40 rpm. Particles were classified along the x-axis.
physical properties used in the simulations and experiments are shown in Table 1. The spring constant k, coefficient of restitution e and friction μ were set to 1000 N/m, 0.9 and 0.3, respectively. These values were chosen after a sensitivity analysis and comparison with experiments from our previous work [11,13,19]. To our knowledge, these values represent the most realistic particle behavior. The damping coefficient η is defined through the restitution coefficient e [34]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk ; 2 ð ln eÞ −π2
η ¼ 2 ln e
ð14Þ
where m is the particle mass. In this study, the restitution coefficient of 0.9 is equivalent to a damping coefficient of 4.46 × 10−3 Ns/m. The value of the simulation time step Δt is important for accurate integration of the motion equations. For the linear spring
model, Tsuji et al. [34] proposed an inequality as an adequate time step Δt: Δt ≤
π 5
rffiffiffiffiffi m : k
ð15Þ
In this study, the critical value for the time step Δt is ~ 4 × 10−5 s which is four times greater than the value of 1 × 10−5 s that we used. One million particles (4.418 kg total mass) were used in the validation tests as shown in Table 2. The rotational speed of the blade was set to 20, 40 and 60 rpm. Granular motion was recorded using a Phantom v9.1 high-speed camera (Vision Research Inc., Wayne, NJ, USA). Experimental data were analyzed using PIV implemented in the DynamicStudio (Dantec Dynamics, Nova Instruments, Woburn, MA, USA) to evaluate the granular flow of particles. Visual data were segmented into 27 by 52 segments and analyzed. Several photographs
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were taken at different times and the velocity of each segment was evaluated using the PIV method. In the simulation case, a similar procedure was also implemented, except that the segment velocity was evaluated by averaging the nearby particle velocities. All simulations were carried out using multi-thread parallelized DEM on a single personal computer. The computational code was developed in our laboratory [35] and was validated in our previous studies [11,13,19]. The time required for the simulation of one full time step are ~0.15, 0.22 and 0.38 s for 400,000, 700,000 and 1,000,000 particle cases, respectively. 3.2. Results and discussion Three validation tests were performed. Velocity distributions of the solid phase because of different mixing speeds were compared qualitatively and quantitatively between the simulation and experiment. The comparison was made by observing the granular flow at the top of the mixing device. Fig. 3 shows the velocity distribution of the solid phase obtained from the simulation and experimental results in Cases 3–1, 3–2 and 3–3. Convective flows were observed in all cases. The tendency to granular flow, namely, the direction and magnitude of the velocity
distribution, agreed well between the simulation and experimental results. In all cases, solid particles near the blades achieved relatively higher velocities. Granular flow increases in a similar manner to the increase in rotational speed. The highest speeds increase from 0.2 to 0.5 m/s with an increase in blade speed from 20 to 60 rpm (note the different scales in Fig. 3). For a quantitative comparison, the average particle speed was measured across x and z directions (lines at Fig. 3) at several discretized positions by averaging one full rotation at each analyzed mixing speed. The average particle speed was chosen since particle movement is mostly one directional. The simulated and experimental values of the average particle speed are shown in Fig. 4. The average particle speeds agree relatively well between the simulations and experiments in all analyzed mixing speeds. The highest average particle speed exists in the middle of the mixer and increases from ~0.08 m/s to 0.25 m/s with increase in mixing speed from 20 rpm to 60 rpm. The average particle speed decreases (~50% of the central value) similarly (almost identical in the simulations) from the center to both sides along the z direction (Fig. 4b). Along the x direction (Fig. 4a) this decrease is much higher and not symmetrical. This effect is connected to the avalanche of particles along the x direction. The highest average particle speed in the center of the mixer can be explained by considering that
Fig. 7. Mixing states at different time (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles at blade speed of 60 rpm. Particles were classified along the x-axis.
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the two sets of blades intersect at that point (Fig. 2) to give more particle movement in one rotation compared with the sides. In summary, qualitative and quantitative agreement between the computed and experimental results for different mixing speeds implies that the DEM/SDF is adequate to simulate granular flow in the ribbon mixer. Convective granular flow could be simulated using a simple linear spring system, and the complex-shaped mixer surface could be modeled by the SDF. Consequently, adequacy of the DEM/SDF was shown through the validation tests. This indicates that the DEM/SDF can simulate detailed granular flow in the ribbon mixer. Therefore, DEM/SDF is applied to investigate solid mixing efficiency in the ribbon mixer. 4. Mixing mechanism in the ribbon mixer Mixing states can be evaluated based on particle positions in the ribbon mixer. To investigate the mixing efficiency of solid particles in the ribbon mixer, DEM simulations were performed by varying the blade speed and particle number. Central shaft speeds were chosen as 20, 40 and 60 rpm and the number of granules was 400,000, 700,000 and 1,000,000. Therefore particle behavior was simulated for nine cases (see Table 2) and 20 s of physical time. The physical properties were the same as those used in the validation tests (Table 1). Particles were distinguished using different colors, namely, black and white, where
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the number of each was equivalent. Simulation cases were compared visually using the colored particle distributions. A visual observation of the mixing behavior reveals the scope of mixing and the different axial coloring of the particles helps to clarify the different mixing behaviors. Lacey's mixing index was used to evaluate these mixing states numerically.
4.1. Visual observations of mixing The effect of the number of particles and blade speed on mixing efficiency was investigated. Fig. 5 shows the mixing states at different time steps for an increasing number of particles whereas the mixing speed was 20 rpm in all calculations (Cases 1–1, 2–1 and 3–1). The same cases are shown in Figs. 6 and 7 for mixing speeds of 40 rpm (Cases 1–2, 2–2 and 3–2) and 60 rpm (Cases 1–3, 2–3 and 3–3), respectively. Better mixing can be achieved using more particles and faster blade speeds. For the 400,000 and 700,000 particles, masses of white and black particles existed even after 20 s of mixing for all speeds. When 1,000,000 particles were used, the black and white particles seemed to achieve a homogenous distribution only after 20 s and at 60 rpm speed. Therefore, the mixing efficiency is influenced by the amount of powder and the blade speed.
Fig. 8. Mixing states for Case 4–3 at different times (0, 5, 10 and 20 s from top down) using particle classification along the x-, y- and z-axes.
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Fig. 9. Mixing states at different times (0, 5, 10 and 20 s from top down) for 400,000, 700,000 and 1,000,000 (from left to right) particles with blade speed of 60 rpm using particle classification along the z-axis.
The effects of initial powder loading were analyzed by classifying particles in different directions. In Figs. 5, 6, and 7 the classification is only along the x-axis. In Fig. 8, classifications along the x-, y- and z-
axes are used for 1 million particles mixed at 60 rpm (Case 3–3). After some time, the y-axis mixing state is similar to the x-axis mixing state. However, z-axis mixing is very different and particles remain unmixed even after a long time. Furthermore, the use of different particle numbers does not generate a good mixture along the z-axis. Fig. 9 shows similar states of a low degree of mixing along the z-axis for all analyzed particle numbers (Cases 1–3, 2–3 and 3–3). It is clear that in this mixer, the diffusion effect in the z-direction is quite small. 4.2. Mixing states
Fig. 10. Mixing indices for different sampling volumes in Case 3–3. Particle classification is along the x-axis.
Although a visual comparison gives a good idea of the mixing scope, it is insufficient. The use of a mixing index allows for a quantitative evaluation of the mixing state. The sample volume is important when analyzing the mixing index [1]. Different sample volumes were tested before the calculations were conducted using the mixing index in Case 3–3 (1 million particles mixing at 60 rpm). The sample volume was expressed as a ratio of sample volume to particle volume. Several ratios ranging from 2 to 1644 were tested. The results are shown in Fig. 10, where it is seen that the mixing index increases with sample volume. Given that almost all particles are sampled, larger samples can have
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Fig. 11. Effect of blade speed on the mixing state (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the x-axis.
larger chunks to achieve the same mixing index as smaller samples. Therefore, the mixing index increases. For subsequent calculations, the ratio was chosen to be 247, which corresponds to a cube with sides of approximately 10 mm. Such a ratio was chosen to contain sufficient number of samples with adequate sample size. If the amount of particles in a sample is less than 70% of the closely packed sphere number, then the sample is discarded. The average number of samples is ~ 1500, 2450 and 3200 for the 400,000, 700,000 and 1,000,000 particle cases, respectively, which give a standard error of measured variance of ~3.7%,
Fig. 12. Effect of amount of powder on the mixing state (blade speeds of 20, 40 and 60 rpm for 400,000, 700,000 and 1,000,000 particles). Particle classification is along the x-axis.
2.9% and 2.6%, respectively. Such values show that our calculated mixing indices are sufficiently accurate to be compared across cases. 4.2.1. Effect of blade speed Fig. 11 shows the change in mixing index for all the simulated cases by grouping cases with the same number of particles. The higher blade
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Fig. 13. Effect of blade speed on the mixing state (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the y-axis.
Fig. 14. Effect of blade speed on the mixing state (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the z-axis.
speeds generate better mixing for all the analyzed particle numbers (cases are compared visually in Figs. 5, 6 and 7). Therefore, convective mixing can be enhanced using a higher mixing speed.
4.2.2. Effect of amount of powder The effect of the amount of powder can be analyzed by grouping the mixing indices with the same blade speed as shown in Fig. 12. Better mixing is achieved by increasing the number of particles. Such an effect was also observed by comparing the particle distribution visually
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Fig. 15. Effect of amount of powder on the mixing state (blade speeds of 20, 40 and 60 rpm for 400,000, 700,000 and 1,000,000 particles). Particle classification is along the z-axis.
(Figs. 5, 6 and 7). Better mixing states in 700,000 and 1,000,000 particle cases indicate that without a sufficient amount of powder, good mixing performance cannot be achieved even after a long time.
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Fig. 16. Mixing index versus number of blade rotations (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the x-axis.
4.2.3. Effect of initial powder location As was done in the visual analysis using different axis classifications, the effect of initial powder loading can be investigated. An analysis of the mixing index shows that mixing in the y-axis direction (Fig. 13) is very similar to mixing in the x-axis direction (Fig. 11). This occurs
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Fig. 18. Average axis-mixing of 400,000, 700,000 and 1,000,000 particles at blade speed of 60 rpm.
of particle distributions along different axes (Figs. 8 and 9). It therefore appears that the mixing of 1,000,000 particles is no longer the best option. This can be seen better by grouping the same blade speed in Fig. 15. The mixing of 1 million particles is slower than that of a smaller number of particles, and this tendency strengthens with increase in blade speed. In addition, the mixing of 700,000 particles is better than that of 400,000 particles. The mixing efficiency with 400,000 particles decreases with an increase in mixing speed compared with that for 700,000 particles. Mixing along the z-axis is extremely low. Thus, an increase in mixing degree along this axis using 700,000 particles may be irrelevant. Such a behavior may also be associated with a different mixing mechanism, namely, shear mixing. Similar results, where an increase in powder reduced the mixing efficiency, were also observed in recent studies of slant cone mixers [27] and industrial twin-screw kneaders [29]. 4.2.4. Mixing per number of blade rotations Another comparison between mixing indices can be achieved by expressing the mixing index versus number of blade rotations and not time. The mixing index versus number of rotations for particles classified along the x-axis and grouping by particle number is shown in Fig. 16. Each group's mixing index is expressed in such a way that they are almost identical. The high mixing rate starts to decline after approximately five blade rotations in all cases. The plateau, which the mixing rate reaches, increases with number of particles. Therefore, only mixing speed reduces the required mixing time. Since particle behavior in the x- and y-directions is similar, similar results can be obtained. However, mixing in the z-direction is different. The mixing index expressed by a number of blade rotations using particle classification along this direction exhibits similar effects (Fig. 17). However, in this case, the mixing index increases linearly in all the analyzed particle numbers. It is unclear when the mixing index reaches a plateau or if the blade speed affects it. Extrapolation of the mixing indices with the lowest analyzed mixing speed to compare it with the highest is unclear since small differences in the inclinations of the mixing lines are within the mixing error. It can be said with certainty, however, that the highest measured mixing indices exist for the 700,000 particle cases. Fig. 17. Mixing index versus number of blade rotations (400,000, 700,000 and 1,000,000 particles mixed at blade speeds of 20, 40 and 60 rpm). Particle classification is along the z-axis.
because these two directions are perpendicular to the rotation axis of the blades. However, mixing along the z-axis (which is the same as the rotation axis) is different. Mixing indices along the z-axis are shown in Fig. 14. It can be seen that the mixing index increases almost linearly with time. These results agree well with the visual comparison
4.2.5. Overall mixing In summary, mixing along the axes perpendicular to the rotation axis is better and increases with increase in speed and number of particles. Mixing in the axial direction is extremely slow, and increases for the 700,000-particle case. The overall mixing behavior can be represented by plotting the average axis-mixing index at 60 rpm with different numbers of particles (Cases 1–3, 2–3 and 3–3) (Fig. 18). The overall mixing of 700,000 particles improved over the 1,000,000 particle mixing after ~ 15 s, whereas the 400,000-particle case produced the
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lowest mixing performance. The main reason why the mixing of 700,000 particles was suggested as the optimal case can be explained by considering the significant increase in mixing of the 700,000 particles in the axial direction compared with that of the 1,000,000 particle mixing. It can be argued that an increase in mixing performance in the axis perpendicular to the axial direction can be associated with convective mixing, whereas a sudden decrease in mixing performance for the 1,000,000 particle in the axial direction shows shearing mixing effects. This may occur because of two separate blades in this direction (see Fig. 2). These blades are inverted, which results in an ineffective convection mixing and an increased effect of shear mixing. It is clear that in such a ribbon mixer, the design of the powder loading is important. 5. Conclusions In this study, the DEM with a wall boundary model based on SDF was used to simulate granular flow in a ribbon mixer. The simulations were validated qualitatively and quantitatively using experiments. The effects of amount of powder, blade speed and initial loading on the degree of mixing were investigated. The following conclusions can be made: a) This study demonstrates that granular flow in a ribbon mixer can be simulated by the DEM/SDF. b) Adequacy of the DEM/SDF was demonstrated through a qualitative and quantitative comparison between the simulated and experimental results. The simulated and experimental velocity distributions and average particle speeds agreed well for all the analyzed mixing speeds. c) An analysis of the simulation data shows that, for the analyzed ribbon mixer, the amount of powder and the blade speed are important factors that affect the mixing quality. The mixture quality increases with an increase in the amount of powder and the blade speed. However, the amount of powder is more important since, without a sufficient amount, a good mixture cannot be achieved even at high mixing speeds. d) The mixing index expressed per number of blade rotations does not depend on the mixing speed. e) An analysis of axis-dependent mixing reveals the importance of the initial location of the powder since particle effects depend on the axis orientation. The mixing rate in the axial direction is extremely low compared with that in the perpendicular directions. f) In the axial direction, optimal mixing can be achieved with 700,000 particles, whereas in the perpendicular directions, such mixing can only be achieved using 1,000,000 particles. Such an effect explains the improved average axis-mixing performance of 700,000 particles. g) A visual evaluation of the mixing behavior agrees well with the mixing index evaluation. This work shows that DEM techniques are starting to reach application at industrial level. A mixing analysis using the DEM/SDF approach can be applied easily to other mixers to provide insightful results.
Nomenclature Roman letters d particle diameter (m) e coefficient of restitution (−) ! F contact force (N) ! FG gravitational force (N) I moment of inertia of a particle (kg m2) k spring constant (N/m) M mixing index (−) m particle mass (kg) N number of particles (−) n sample size (−) R particle radius (m) ! r ðtÞ position vector of a particle (m)
! T t Δt y y
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torque (N m) time (s) time step of the simulation (s) fraction of chosen particles in a sample (−) average fraction of chosen particles in all samples (−)
Greek letters ! δ displacement (m) η coefficient of damping (Ns/m) μ coefficient of friction (−) ρ particle density (kg/m3) σ standard deviation (−) φ signed distance function (m) ! ω particle angular velocity (rad/s) Subscripts i particle index j index of a particle interacting with particle i k wall index N normal component T tangential component 0 fully segregated state r fully mixed state rand random sample A type-A particles B type-B particles Superscripts i sample index P–P particle–particle interaction P–W particle–wall interaction Operator ∇ gradient operator (m−1) Acknowledgments Part of this study was supported financially by MAKINO MFG. Co., Ltd. The authors are grateful to Dr. Yusuke Shigeto for his kind support. References [1] J. Bridgwater, Mixing of powders and granular materials by mechanical means—a perspective, Particuology 10 (2012) 397–427, http://dx.doi.org/10.1016/j.partic. 2012.06.002. [2] S. Masiuk, Power consumption, mixing time and attrition action for solid mixing in a ribbon mixer, Powder Technol. 51 (1987) 217–229, http://dx.doi.org/10.1016/00325910(87)80022-0. [3] S. Masiuk, H. Łacki, F. Strek, Power consumption and mixing times for liquid mixing in a ribbon mixer, Chem. Eng. J. 48 (1992) 135–140, http://dx.doi.org/10.1016/ 0300-9467(92)85015-2. [4] J. Thyn, K. Duffek, Powder mixing in a horizontal batch mixer, Powder Technol. 15 (1975) 193–197, http://dx.doi.org/10.1016/0032-5910(76)80048-4. [5] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulate systems: a review of major applications and findings, Chem. Eng. Sci. 63 (2008) 5728–5770, http://dx.doi.org/10.1016/j.ces.2008.08.006. [6] A.N. Huang, H.P. Kuo, Developments in the tools for the investigation of mixing in particulate systems — a review, Adv. Powder Technol. 25 (2014) 163–173, http:// dx.doi.org/10.1016/j.apt.2013.10.007. [7] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1979) 47–65. [8] F. Bertrand, L.A. Leclaire, G. Levecque, DEM-based models for the mixing of granular materials, Chem. Eng. Sci. 60 (2005) 2517–2531, http://dx.doi.org/10.1016/j.ces. 2004.11.048. [9] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulate systems: theoretical developments, Chem. Eng. Sci. 62 (2007) 3378–3396, http://dx. doi.org/10.1016/j.ces.2006.12.089. [10] X. Sun, M. Sakai, Y. Yamada, Three-dimensional simulation of a solid–liquid flow by the DEM–SPH method, J. Comput. Phys. 248 (2013) 147–176, http://dx.doi.org/10. 1016/j.jcp.2013.04.019. [11] M. Sakai, Y. Shigeto, J. Xiong, T. Saito, X. Sun, S. Koshizuka, et al., Lagrangian– Lagrangian modeling for a solid–liquid flow in a cylindrical tank, Chem. Eng. J. 200–202 (2012) 663–672, http://dx.doi.org/10.1016/j.cej.2012.06.080.
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