Powder Technology 217 (2012) 148–156

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Numerical modeling of fluid and particle behaviors in impact pulverizer
Hirohisa Takeuchi, Hideya Nakamura, Tomohiro Iwasaki, Satoru Watano ⁎
Department of Chemical Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

a r t i c l e

i n f o

a b s t r a c t
Numerical modeling of fluid flow and individual particle motion in an impact pulverizer was conducted using a Computational Fluid Dynamics (CFD)–Discrete Phase Model (DPM) coupling model. The impact pulverizer used is a dry system. Its grinding chamber consists of high-speed rotating hammers and a static concavo–convex stator. First of all, calculated results of fluid pressure in the grinding chamber were compared with the experimental ones, showing the both results were in good agreement. The fluid flow in the grinding chamber indicated that the fluid mainly swirled in the direction of the hammer rotation. The fluid flow pattern in the concaves of the stator was also swirling flow, while its velocity was much lower than that in the outside of the concaves. Analyses of the particle motion suggested that the particles were accelerated by the fluid drag force caused by the rotating hammers but not by the impact force from the hammers, resulting in impacts with the static stator at the velocity 1.4 times higher than the tip speed of hammers. The velocities and frequencies of the particle impacts with the walls of the grinding chamber were also investigated under various rotor speeds and particle sizes. As the particle size was smaller, the particle impacts with the stator were not likely to occur, and a decrease of the impact velocity was observed despite the same rotor speed. It was because the particle velocity decreased before impact with the stator due to the fluid resistance force acting on the particle in the concave. The fluid resistance force was proportional to the square of the particle size. By constant, an inertia force of the particle was proportional to the particle mass, which was equivalent to the cube of the particle size. Accordingly, the velocity of particle impact with the stator was lower when particle size was smaller. These calculated results implied that the limitations of ground particle size existed because of the decrease in the frequency and velocity when particle size was smaller. © 2011 Elsevier B.V. All rights reserved.

Article history: Received 3 May 2011 Received in revised form 12 August 2011 Accepted 13 October 2011 Available online 20 October 2011

Keywords: Impact pulverizer Numerical modeling Grinding CFD DPM Two-phase flow

1. Introduction An impact pulverizer is one of the most important powder handling processes, which can rapidly reduce particle size down to a few 10 μm under dry condition by an interaction between highspeed rotating hammers and a static concavo–convex stator. This is a ubiquitous powder handling processor in many industrial sectors, such as pharmaceutical, food, agrichemical, and so on. Particles in an impact pulverizer are carried by fluid flow caused by the rotating hammers and then impact with walls of the pulverizer, resulting in grinding of the particles. Thus, physical properties of the ground particles are greatly affected by the fluid flow in the impact pulverizer. This means that understanding of fluid flow and impact mechanism of individual particles in the impact pulverizer is very important for the better design and control. So far, dynamics of the fluid and particle in the impact pulverizer has been theoretically analyzed. For example, Austin [1] proposed a theoretical model for predicting the particle size distribution of

⁎ Corresponding author. Tel.: + 81 72 254 9305; fax: + 81 72 254 9217. E-mail address: watano@chemeng.osakafu-u.ac.jp (S. Watano). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.10.021

ground particles in an impact pulverizer using the net milling power. The net milling power was estimated based on an impact force between the rotating hammers and a particle-suspending fluid in the impact pulverizer. Although this theoretical model is somewhat useful for estimating the particle size distribution, the fluid flow and individual particle motion in the impact pulverizer could not be revealed. Additionally, this model cannot be applied to impact pulverizers having complicated geometries. Recently, numerical simulations have become powerful tools for the analysis of fluid flow and individual particle motion in various types of grinding processes, such as tumbling mills and stirred media mills [2]. A Computational Fluid Dynamics (CFD), in which governing equations of fluid motion are numerically solved, has been a major technique for modeling of fluid flow [3]. By using the CFD, fluid flows in various types of grinding processes having complicated geometries can be simulated [4–7]. The CFD can be also applied to gas–solid two-phase systems by coupling with other simulation models. A coupling model of the CFD with a discrete element method (CFD–DEM coupling model [8,9]), in which the fluid flow as well as individual particle motion can be simulated, has been widely used in many types of powder handling processes such as fluidized beds and pneumatic conveyings. This model has also been used for analyses of the Isa mill [10,11] and a fluid energy mill [12]. A coupling model of the CFD with a discrete phase model (CFD–DPM

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coupling model) is another technique for modeling of powder handling processes with gas–solid flows [13,14]. The particles are modeled as point mass in this model, thus the particle-to-particle impacts are not taken into account. This leads to lower computational load of this model as compared with that of the CFD–DEM coupling model. Therefore, the CFD–DPM model is suitable for modeling of impact pulverizers, since the solid flow can be regarded as a dilute phase [15]. Using this model, the fluid flow and motion of raw particulate materials and their ground products in impact pulverizers have been simulated [15]. However, this simulated pulverizer was composed of a fan rotor and a flat stator of simple geometry, in which there is little interaction between the rotor and stator. Most of the impact pulverizers widely used for fine grinding in industrial sectors are composed of a rotor and a stator of complicated geometry, in which there is strong interaction between the rotor and concave–convex stator. Therefore, the fluid flow as well as the velocity and frequency of the particle impact in the impact pulverizer having the rotor and concave–convex stator of complicated geometries is strongly required to be clarified. In this paper, the fluid flow and individual particle motion were numerically analyzed using a CFD–DPM coupling model to investigate the mechanism of particle impact in an impact pulverizer. In order to confirm validity of the proposed numerical model, calculated results of the fluid pressure was compared with experimental ones. Pressure and velocity of the fluid in the impact pulverizer were then analyzed. The particle motion was numerically calculated and compared with the actual one observed using a high-speed video camera. Also, the velocities and frequencies of the particle-to-wall impact were analyzed at various particle diameters and rotor speeds, and mechanism of particle-to-wall impact was discussed. 2. Impact pulverizer Fig. 1 shows a schematic diagram of impact pulverizer (Hammer mill, LM-05, Fuji Paudal Co., Ltd.) used in this study. The impact pulverizer consists of a high-speed rotating rotor having eight hammers with dimensions of 15 mm in width × 8 mm in height × 8 mm in depth, a grinding chamber with a diameter of 119 mm and a depth of 30 mm, a concavo–convex-shaped stator on the chamber wall, a

classification screen with opening size of 1 mm in diameter, and a collection pot of products. A minimum gap between the hammer tip and the stator protuberance is 0.5 mm. The maximum rotor speed is 267 rps, i.e., the maximum tip speed of hammers is 105.7 m s − 1. The classification screen is installed between the grinding chamber and the collection pot. A raw particulate material is continuously fed into the center of the chamber and ground between the high-speed rotating hammer and the static stator. The ground particles which pass through the screen are collected in the pot as the product. 3. Numerical modeling The fluid flow and individual particle motion in the impact pulverizer were simulated using a Computational Fluid Dynamics (CFD)– Discrete Phase Model (DPM) coupling model. The fluid flow was calculated using the CFD. The fluid flow in the impact pulverizer is extremely high-speed turbulent flow caused by high-speed rotating hammers. Therefore, in this CFD simulation, the fluid was assumed to be a viscous and compressible fluid, and the fluid flow was considered as turbulence and unsteady flow. The fluid flow was calculated by solving the equation of continuity, the Reynolds-averaged Navier– Stokes (RANS) equation, the equation of energy, and the equation of state of ideal gas. The governing equations are as follows:Equation of continuity

∂ ρ þ ∇⋅ðρuÞ ¼ 0 ∂t Equation of motion (RANS equation)

ð1Þ

∂ ðρuÞ þ ∇⋅ðρuuÞ ∂t     2 Z  ′ ′ T þ ρg ð2Þ ¼ −∇p−∇⋅ −μ ∇u þ ð∇uÞ þ μδð∇⋅uÞ þ ρ u u 3

Raw materials
1
2 1

2
3 4

9 7

119 mm

6
5
8

Products

Side view
1. Feeder 4. Rotor 7. Grinding chamber

Front view
2. Front cover 5. Classification screen 8. Collection pot 3. Stator 6. Bag filter 9. Motor

Fig. 1. Schematic diagram of impact pulverizer used in this study.

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Equation of energy

  1 ∂ 1 2 2 ^ ^ ρjuj þ ρU þ ∇⋅ ρjuj þ ρU u 2 ∂t 2 2 ¼ −∇⋅pu þ keff ∇ T−ρðu⋅∇jg jhÞ
 

ð3Þ

Equation of state of ideal gas ρ¼

p0 þ p ðRm =Mw ÞT

ð4Þ

where ρ u, u′, p, μ, g, and δ are fluid density, fluid velocity, fluctuating velocity of fluid, fluid pressure, fluid viscosity, gravity acceleration, and the Kronecker delta, respectively. Û, keff, T, and h indicate internal energy of fluid, effective heat conductivity of fluid, fluid temperature, and height from reference plane, respectively. p0, Rm, and Mw are standard pressure, molar gas constant, and molar weight of the fluid, respectively. The Boussinesq approximation [16] was applied to solve the Reynolds’ stress term ρ u′ u′ in Eq. (2), which was calculated by the following equation:   2 Z 2 ′ ′ T ρ u u ¼ −μ t ∇u þ ð∇uÞ þ μ t δð∇⋅uÞ þ ρkδ 3 3

Z

where μhg and uvg are horizontal and vertical fluid velocities, respectively. xi and ui are fluid length and fluid velocity to the direction of i, respectively. These constants were determined according to a previous study [17]. The modified k−ε turbulent model [17] used can simulate a swirling flow more accurately than a standard k−ε turbulent model [18], because Cμ is calculated at each computational cell in each time step in the modified k−ε turbulent model, although Cμ is treated as a constant in the standard k−ε turbulent model. The fluid pressure and velocity were calculated by the SIMPLE algorithm [19]. The fluid temperature was calculated from the equation of energy expressed by Eq. (3). The fluid density was then calculated from the equation of state of ideal gas expressed by Eq. (4) using the calculated fluid temperature. The individual particle motion was calculated using the DPM. In the DPM, the particles are treated as point mass, and the particleto-particle impacts are not taken into account. The DPM can be valid when the solid volume fraction is less than 10 vol.% [15]. It was reported that the solid volume fraction in an impact pulverizer is less than 10 vol.% [15]. Therefore, the DPM was used in this study for analyzing the individual particle behavior in the impact pulverizer. The individual particle motion was calculated by solving the equation of motion as follows: ρp

ð5Þ

where μt and k indicate the eddy viscosity coefficient and turbulent kinetic energy, respectively. The modified k−ε turbulent model [17] was used in this study. In this model, the eddy viscosity coefficient μt, the transport equations of the turbulent kinetic energy k, and the turbulent energy dissipation rate ε are expressed by the following equations: μ t ¼ ρC μ

  dup ¼ F d þ g ρp −ρ dt

ð14Þ

The fluid drag force, Fd is written as Fd ¼

 18μ C d Rep  ρp u−up 2 24 ρp dp

ð15Þ

k ε

2

ð6Þ

Transport equation of turbulent kinetic energy

Rep ¼

    ρdp up −u μ

ð16Þ

 ∂ ∂  ∂ ρkuj ¼ ðρkÞ þ ∂xj ∂xj ∂t

"  # ∂k μ þ Gk þ Gb þ ρε þ Y m μþ t σ k ∂xj

ð7Þ

Transport equation of turbulent energy dissipation rate

 ∂ ∂  ∂ ρεuj ¼ ðρε Þ þ ∂xj ∂xj ∂t

"  # 2 ε μ t ∂ε pffiffiffiffiffiffi þ ρC 1 Sε þ ρC 2 μþ σ ε ∂xj k þ νε ε þ C 1ε C 3ε Gb k

ð8Þ

where, Gk, Gb, Ym, and ν are terms indicating generation of turbulent kinetic energy due to the mean velocity gradients, generation of turbulent kinetic energy due to buoyancy, contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, and kinematic viscosity, respectively. xj and μj are fluid length and fluid velocity to the direction of j, respectively. C2 and C1ε are constants. σk and σε are turbulent Prandtl numbers for k and ε, respectively. C1, C3ε, and S are calculated from the following equations:

 C 1 ¼ max 0:43; η¼S

η ηþ5



ð9Þ

k ε

where up, ρp, dp, and Rep are particle velocity, particle density, particle diameter, and particle Reynolds number, respectively. The coefficient of fluid drag Cd is obtained from the empirical correlation of Haider and Levenspiel [20]. A lactose, which is a common pharmaceutical excipient, was used as the model particle in this analysis. The size distribution of model particles is treated as mono-disperse. The coefficient of restitution for particle-to-wall impact was set as 0.3 [21]. The breakage and size reduction of the particles were not considered in this simulation because we mainly focused on a particle behavior in the grinding chamber. Calculation conditions used in this simulation are listed in Table 1. Three-dimensional geometry for the simulated impact pulverizer was created using the GAMBIT (Ver. 2.4.6, Fluent Inc.) based on the CAD data. Actual appearance and geometry of the simulated pulverizer are shown in Fig. 2a and b. Actual appearance and fluid-grids of the grinding chamber are also indicated in Fig. 2c and d. The fluid-grids mostly consist of hexahedral elements in order to improve speed and stability of the calculation. The total number of grids is 490,436. The fluid-grids mainly consist of three regions: (1) region of the front cover with the inlet boundary of fluid, (2) region of the grinding chamber with the rotating hammer, and (3) region of the stator and collecting pot with the outlet boundary of fluid. The fluid-grids in the region (2) were rotated with the motion of hammer, although the fluid-grids in
Table 1 Calculation conditions. Fluid density (101.325 kPa, 300 K) Fluid viscosity (101.325 kPa, 300 K) Inlet pressure Outlet pressure Time step Particle density Restitution coefficient of particle-to-wall 1.225 1.789 × 10− 5 101.325 101.325 8.333 × 10− 6 [kg m− 3] [Pa s] [kPa] [kPa] [s] [kg m− 3] [–]

ð10Þ ð11Þ

qffiffiffiffiffiffiffiffiffiffiffiffi S ¼ 2Sij Sji Sij ¼

C 3ε

1 ∂uj ∂ui þ 2 ∂xi ∂xj   u   hg  ¼ tanh  uvg 

!

ð12Þ

ð13Þ

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Feed

Collect

a

b

investigated. A schematic diagram of the experimental set-up is shown in Fig. 3. Geometry and scale of the impact pulverizer used in this experiment were the same as that used in the numerical simulation. A pressure sensor (AP-12S, Keyence Corp.) was used for measurement of fluid pressure at the inside of the grinding chamber. A pressure tap was installed at the top of the grinding chamber. The front cover is made of a transparent acrylic plastic, which allows to observe the particle motion. The particle motion was visualized using a high-speed video camera (Phantom V710, Vision Research Inc.). Recording speed of the high-speed video camera was set 100,000 fps. Spherical particles made of sucrose (Nonpareil-101, Freund Corp.) were used as the experimental model particles. The size range of the model particles was between 500 and 710 μm. 5. Results and discussion 5.1. Validation of simulation model In order to validate our proposed model, the calculated results were compared with the experimental ones. Previous researches [15,23] validated their proposed simulation models by comparison between the calculated fluid pressures in grinding processes with the experimental one. In this study, the calculated fluid pressure in the grinding chamber was compared with the experimentally measured one as well. The time averaged fluid pressure as a function of the rotor speed is shown in Fig. 4. The location of measurement sensor of fluid pressure in the experiment was the same as in the numerical simulation. The calculated results showed good agreement with the experimental ones. This implies that validity of the simulation model was quantitatively confirmed. The fluid pressure slightly increased with an increase in the rotor speed, because the fluid was compressed by the hammers at circumference of the grinding chamber. 5.2. Fluid pressure in grinding chamber Fig. 5 shows calculated static fluid pressure distribution in the grinding chamber at a cross section of the rotational axis. The data was obtained after the outlet mass flow rate of the fluid from the

c

d

Fig. 2. Actual appearance and fluid-grids of impact pulverizer. (a) Actual overall view of pulverizer. (b) Numerical geometry of pulverizer. (c) Actual view of grinding chamber. (d) Fluid-grids of grinding chamber.

the region (1) and (3) were stationary. The interface between the rotating grids and the stationary grids was treated using the sliding mesh model [22]. 4. Experimental In order to validate our proposed numerical model by comparing between the calculated and experimental results, the fluid pressure and particle motion in the impact pulverizer were experimentally

6
5
4

8
7
9

1

2
3

10

1. Motor 4. Front cover 7. Illumination lamp 10. Data analysis system

2. Bag filter 5.Pressure tap 8. Light source

3. Collection pot 6.Pressure sensor 9. High speed video camera

Fig. 3. Experimental set-up for measurement of fluid pressure and observation of particle motion.

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115
110
105
100
95
90 100

Fluid pressure [kPa]

Calculated Experimental

pattern in concaves of the stator greatly differ from those in other parts. In the concaves, the fluid velocity was much lower than the tip speed of the hammers. The swirling flows inside the concaves were generated by passing of the high-speed hammers over the concaves. 5.4. Particle behavior in grinding chamber Fig. 8 indicates a calculated result of individual particle motion in the grinding chamber. Particles were continuously fed into the chamber with rotating of the hammer. In Fig. 8, particles are colored according to its velocity, and the size of the particle is enlarged by 3 times in order to clearly visualize. The number of the simulated particles in Fig. 8 was approximately 45,000. The number of the particles was determined based on a preliminary study, in which it was confirmed that when the number of the particles is more than 4000, the velocity distribution of particle impact with wall was unchanged. From the results, the particles were initially transported from center to circumference of the grinding chamber, and then accelerated by the fluid drag force, caused by the rotating hammers. The particles mostly exist at the circumference because of the centrifugal force acting on the particles. The calculated solid volume fraction was less than 3.4 vol.%. Therefore, it was confirmed that influence of particle phase to gas phase can be ignored. Fig. 9 describes a typical particle motion in a concave of the stator when a hammer passed over a concave. Distributions of the fluid pressure and fluid velocity are also indicated in the background. Time interval between snapshots was 33.3 μs. It should be noted that particles in front of the hammer were accelerated only by the fluid drag force but not by the impact with the hammer (Fig. 9a). The particles then moved into a concave of the stator with a velocity higher than the tip speed of hammers (Fig. 9b). The particles impacted with the inner wall of the concave (Fig. 9c), and then the particles rebounded (Fig. 9d). The fluid pressure in the vicinity of the impact location with the particles was remarkably compressed. Fig. 10 shows the actual particle motion in a concave observed by the high-speed video camera. The time interval between recorded images was 40 μs. The particle motion in the concave was similar to the calculated ones. The observed particle impact location on the inner wall of the concave was very similar to the calculated results. The actual particle velocity obtained from the experimental result was 41.7 m s − 1. The particle velocity in the calculated results at the same rotor speed showed from 40 to 70 m s − 1. The particle velocity in the experimental results thus showed good agreement with the calculated one. From these results, it was found that the particles were accelerated only by the fluid drag force without the contact with the hammer, and then particles impact with the stator at a velocity higher than the tip speed of hammers. 5.5. Effects of operating parameters on velocity and frequency of particle impact The rotor speed and the particle size of raw materials are critical operating parameters in the impact pulverizers [24]. Thus, the velocities and frequencies of particle impact with the entire wall of the grinding chamber were numerically analyzed at various rotor speeds and particle sizes. Especially, we investigated the particle impact with two kinds of walls: (i) front walls of hammers and (ii) inner walls of concaves of the stator, as shown in Fig. 11. We used 4000 particles for analyzing the effects of operating parameters. It was preliminary confirmed that 4000 particles are enough to analyze the particle impact. Fig. 11 shows velocity distributions of the particle impact with the walls of hammers and concaves at different rotor speeds. The particle size was 250 μm. The axis of ordinate and abscissas indicates the number of particle-to-wall impacts per second and the impact velocity of particle-to-wall in the normal direction, respectively. The velocity

150

200

250

300

Rotor speed,R [rps]
Fig. 4. Comparison of fluid pressures obtained from calculated and experimental results in impact pulverizer.

exhaust reached almost constant. The fluid pressure at circumference of the grinding chamber was higher than in other areas. Especially, the fluid pressure in the concave of the stator when the hammer tips approached to the concave was extremely high. This means that the fluid in the concave was strongly compressed by an interaction between a high-speed rotating hammer and a static concave–convex stator. By contrast, the fluid pressure near the center of the grinding chamber was lower than the standard atmospheric pressure (101.325 kPa). This negative pressure causes the intake flow into the grinding chamber. The radial gradient of fluid pressure from center to circumference of the grinding chamber is formed by centrifugal force generated by rotational motion of the fluid. Fig. 6 indicates distributions of the static fluid pressure at the top of grinding chamber under various rotor speeds, R. At higher rotor speeds, the fluid was more compressed. Variation of the fluid pressure in the concaves generated when the hammer tips passed over the concave also increased with an increase in the rotor speed. 5.3. Fluid velocity in grinding chamber Fig. 7 shows a calculated fluid velocity distribution in the grinding chamber at the rotor speed, R of 267 rps. The fluid in the grinding chamber mainly swirled around the rotational axis of the hammers. The fluid velocity at the circumference of the grinding chamber was higher than those in other areas. Especially, the fluid velocity in front of the hammers was higher than the tip speed by 3%. This is because that the fluid in front of the hammers is strongly compressed by high-speed motion of the hammers. The fluid velocity and flow

Static pressure [kPa]
107.9 105.7 103.5 101.3 99.1

R = 267 rps
Fig. 5. Pressure distribution in grinding chamber.

H. Takeuchi et al. / Powder Technology 217 (2012) 148–156

153

R = 267 rps

Static pressure [kPa]
108
R = 200 rps

106 104 101 99.5

R = 133 rps

Fig. 6. Fluid pressure distributions at top of grinding chamber in various rotor speeds.

distribution of the particle impact was sampled for 3.75 ms computing time. The computing time was determined from a preliminary study. According to the preliminary study, the velocity distribution of the particle impact was unchanged when the computing time is longer than 2.36 ms. The impact velocities less than 5 m s− 1 were not shown in Fig. 11, because the impacts with such low impact velocity cannot contribute to the grinding of particles very much. The results indicate that velocity distributions of the particle impact were bimodal at any rotor speeds. The maximum impact velocities of the particle in any rotor speeds were higher than the tip speed of hammers by 1.4 times. The maximum impact velocity and a width of the distribution increased with an increase in the rotor speed. Grand and Kalman [25] reported that in the experiment the size of ground particles decreases and width of the particle size distribution of the ground particles becomes broader under hi rotor speeds. Therefore, the calculated results reflect characteristics of the experimental ground products. The bimodal impact velocity distributions of the particle impact in the impact pulverizer can be classified into the two-types, i.e., high-speed impact and lowspeed impact. In the region of high-speed impact, most of the particles impacted with the stator. The number of impacts with the hammers

was very small as compared with the one with the entire wall of the grinding chamber. Therefore, the particle impact at a velocity higher than the tip speed of hammers mainly occurred on the stator. This implies that the particles in the impact pulverizer are mostly broken by the impact with the stator. Fig. 12 shows number fraction of particle-to-stator impacts in different particle sizes. The rotor speed was 267 rps. The sizes of particles used were 250, 100, 50 and 25 μm. The axis of ordinate indicates the number fraction of the particle-to-stator impact in all impacts with the entire wall. The fraction of low-speed impact (lower than 70 m s − 1) was not affected by the particle size. However, the fraction of particle-to-stator impact at a velocity faster than the tip speed of hammers (105.7 m s − 1) reduced in smaller particle sizes. Generally, a smaller particle moving along with a streamline of fluid flow is more difficult to deviate from the streamline, because an inertial force of a particle decreases with a decrease in size of the particle. The calculated results show that the number fraction of particles, which are deviated from the swirling streamline in the grinding chamber and move into the concaves, decreases when the particle size is smaller.

Fluid velocity [m/s]
120 90 60 30 0

Tip speed of hammer = 105.7 m s-1

R = 267 rps
Fig. 7. Fluid velocity distribution in grinding chamber.

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Particle velocity [m s-1]
0 40 80 120 160

force when a hammer approached the particles. The particle velocity then began to slightly decrease at 0.007 m at which particles moved into a concave of the stator. Before moving into the concave, the particles had much higher velocity than the fluid velocity in the concave. Therefore, the particles are subjected to the fluid resistance force in the concave, leading to decrease of its particle velocity. The particle velocity in the concaves more decreases when particle size was smaller. Generally, the fluid resistance force acting on a single sphere increases with the square of the particle size as indicated from Eq. (15). By constant, an inertia force of the particle is proportional to the particle mass, which is the cube of the particle size. Accordingly, the velocity of particle impact is lower when particle size is smaller. These calculated results show that the frequency and velocity of the particle impact with the stator decrease with a decrease in the size of the particles. This implies that the limitations of ground particle size exist because of the decrease of the frequency and velocity when particle size is smaller. It is concluded that fluid flow is important factor for the velocity and frequency, which determines performance of the impact pulverizers.

6. Conclusions

R = 267 rps

Tip speed of hammer = 105.7 m s-1
Fig. 8. Calculated result of particle behavior in grinding chamber. (Size of the particles is enlarged by 3 times.).

Fig. 13 shows temporal changes in particle velocity before the particle impact with the stator under various particle sizes. The particles in red circles of snapshots from I to IV correspond to I to IV in the graph. The graph indicates the number averaged particle velocity before the particle impact with the stator at a velocity higher than 70 m s − 1. In the 250 μm, the particle velocity began to increase at 0.017 m, because the particles were accelerated by the fluid drag

The fluid flow and individual particle motion in an impact pulverizer were numerically analyzed using a CFD–DPM coupling model. First of all, calculated fluid pressure in the grinding chamber was compared with the experimental one, and both results showed good agreement. It was found that the fluid in the grinding chamber mainly swirled in the direction of hammer rotation, and the fluid velocity in front of the hammers was higher than the tip speed of hammers. It was also found that the fluid in the concaves of the stator swirled inside the concaves at a velocity much lower than the tip speed of hammers. The calculated results of particle motion showed similar behavior to the actual motion of a particle, which was observed by a high-speed video camera. The velocities and frequencies of particle impact with the entire wall of the grinding chamber were also analyzed under various rotor speeds and particle sizes. It was shown that a maximum impact velocity of the par-

Fluid pressure Fluid and particle [kPa] velocity [m s-1]
108 106 104 101 99
160 120 80 40 0

105.7 m s-1

a

b

c

d

Fig. 9. Calculated results of particle behavior around hammer tip and stator. (R = 267 rps, Tip speed of hammer = 105.7 m s− 1, Size of the particles is enlarged by 3 times.).

Particle

1
Inner wall of Concave

2

3

4

Fig. 10. Experimentally observed impact behavior of a particle. (R = 133 rps).

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Number fraction of particle-to-stator impacts [–]

(ii)Stator

0.04

Tip speed of hammer

0.03

0.02

R = 267 rps Particle diameter; 250 µm 100 µm 50 µm 25 µm

0.01

(i)Hammer

0

0

50

100

150

Impact velocity of particles [m

s-1]

Fig. 12. Number fraction of particle-to-stator impacts in different particle sizes.

1E+6
0.8E+6
0.6E+6
0.4E+6
0.2E+6

R = 133 rps, Dp = 250 µm
Tip speed of hammer

Entire wall Stator Hammer

impact mechanism in an impact pulverizer was clarified via the numerical simulation using CFD–DPM coupling model. These results can lead to better design and control of impact pulverizers. Nomenclature C1ε Constant in Eq. (8) [–] C2 Constant in Eq. (8) [–] Cd Coefficient of fluid drag [–] Cμ Coefficient of eddy viscosity model [–] dp Particle diameter [m] Fd Drag force [N m − 3] g Gravity acceleration [m s − 2] Gb Term indicating generation of turbulence kinetic energy due to buoyancy [kg m − 1 s − 3] Gk Term indicating generation of turbulent kinetic energy due to mean velocity gradients [kg m − 1 s − 3] h Height from reference plane [m] k Turbulent kinetic energy [m 2 s − 2] keff Effective thermal conductivity [W m − 1 K − 1] Mw Molar weight of fluid [kg mol − 1] p Fluid pressure [Pa] p0 Standard fluid pressure [Pa] R Rotor speed [rps] Rm Molar gas constant [J mol − 1 K − 1] Rep Particle Reynolds number [–] S Modulus of mean rate-of-strain tensor [s − 1] t Time [s] T Fluid temperature [K] Û Internal energy [J] u Fluid velocity [m s − 1] u′ Fluctuant fluid velocity [m s − 1] ui Fluid velocity to direction of i (i = x, y, z) [m s − 1] uj Fluid velocity to direction of j (j = x, y, z) [m s − 1] up Particle velocity [m s − 1] uhg Fluid velocity to direction of horizontal to gravity [m s − 1] uvg Fluid velocity to direction of vertical to gravity [m s − 1] xi Fluid length to direction of i (i = x, y, z) [m] xj Fluid length to direction of j (j = x, y, z) [m] Ym Term indicating contribution of fluctuating dilatation in compressible turbulence to overall dissipation rate [kg m − 1 s− 3]

0

0

40

80

120

160

Number of particle impacts [s-1]

1E+6
0.8E+6
0.6E+6
0.4E+6
0.2E+6

R = 200 rps, Dp = 250 µm
Tip speed of hammer

0 0

40

80

120

160

1.8E+6

R = 267 rps, Dp = 250 µm
Tip speed of hammer

0.8E+6
0.6E+6
0.4E+6
0.2E+6

0

0

40

80

120

160

Impact velocity of particles [m s-1]
Fig. 11. Impact velocity distribution of particles on inner walls of the grinding chamber.

ticle increases, and a width of the impact velocity distributions become broader with an increase in the rotor speed. As the particle size is smaller, it was also found that the particle impacts with the stator were not likely to occur, and a decrease of the impact velocity was observed despite the same rotor speed. The calculated results implied that the limitations of ground particle size existed because of the decrease of the frequency and velocity when particle size was smaller. Consequently, the particle

Greek letters δ Kronecker delta [–] ε Turbulent energy dissipation rate [m 2 s − 3] μ Fluid viscosity [Pa s] μt Eddy viscosity coefficient [Pa s] v Kinematic viscosity [m 2 s − 1] ρ Fluid density [kg m − 3]

156

H. Takeuchi et al. / Powder Technology 217 (2012) 148–156

Impact point

I
105 .7 m s -1

Particle diameter 250 mm
Particle velocity [m s-1]

II

160 120 80

III

40 0
0.00 8m

IV

IV

III

II

I

140

Particle velocity [m s-1]

120
100
80
60
40
20
0
0
Tip speed of hammer

R = 267 rps
Particle diameter; 250 µm 100 µm 50 µm 25 µm

0.005

0.01

0.015

Distance from particle to impact point [m]
Fig. 13. Temporal changes in particle velocity before particle impact under various particle sizes. Particles in red circles of snapshots correspond to I to IV. (Size of the particles is enlarged by 3 times.).

ρp σk σε

Particle density [kg m − 3] Turbulent Prandtl numbers for k [–] Turbulent Prandtl numbers for ε [–]

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