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Combined effects of internal friction and bed height on the Brazil-nut problem in a shaker
Powder Technology 253 (2014) 561–567
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Combined effects of internal friction and bed height on the Brazil-nut problem in a shaker Chun-Chung Liao, Shu-San Hsiau ⁎, Chi-Sou Wu Department of Mechanical Engineering, National Central University, No. 300, Jhongda Road, Jhongli 32001, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 26 August 2013 Received in revised form 2 December 2013 Accepted 14 December 2013 Available online 21 December 2013 Keywords: Penetration length Friction drag force Granular material Brazil-nut problem Vibration bed
a b s t r a c t We experimentally studied the influence of an intruder's friction coefficient and bed-filling height on the Brazil-nut effect in a quasi-2D vertical vibration granular bed. The motion of intruder was successfully measured using a high-speed camera and the rising time of intruder was determined by using a particle tracking method with the help of image processing technology. The results show that an intruder's friction coefficient and filling bed height play significant roles in the rise dynamics. The results also show that the rise time increases when the intruder's friction coefficient increases, which is reduced when the filling bed height decreases. Penetration length and friction drag force were also determined in this study. The penetration length was reduced and the friction drag force was enhanced with the increase of an intruder's friction coefficient and bed height. Additionally, the variation between the rise times of the smooth and rough intruders was not significant with the lower bed-filling heights. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Granular materials may flow like a liquid or a gas, or behave like a solid. These three motion states may occur simultaneously or exist individually in granular systems [1]. Granular materials are ubiquitous in nature and exhibit a wide range of nontrivial rheological behavior. The rheological behavior of granular materials is complex, but an understanding of granular materials is imperative in many industries, such as the pharmaceutical industry, food storage and transportation, polymer manufacturing, cement manufacturing, and metallurgy. Furthermore, an understanding of granular materials can also be applied to environmental problems such as debris flows and landslides. The segregation phenomenon of granular materials is a critical issue in industrial processes, and this fascinating phenomenon also occurs in vibrated granular beds. The so-called “Brazil-nut problem,” which involves an exceptionally large particle immersed with smaller granular materials in a container that vibrated using a shaker, has been widely studied in recent decades [2–16]. It is well-known that segregation can occur in binary mixtures that have components of different sizes, densities, surface roughnesses, and restitution coefficients. Understanding the segregation mechanisms of granular material is a key scientific and technological challenge to both engineers and scientists. Granular segregation induced by internal friction is a central issue in many industries. Srebro and Levine [14] observed segregation at compactivities in binary mixtures of grains exhibiting differing frictional properties. Plantard et al. [13] observed the occurrence of frictioninduced segregation in a granular slurry shear system, finding that ⁎ Corresponding author. Tel.: +886 3 426 7341; fax: +886 3 425 4501. E-mail address: [email protected] (S.-S. Hsiau). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.12.031
rough circular particles behave similarly to smooth but larger circular particles. Ulrich et al. [16] experimentally demonstrated the transition from the reverse Brazil-nut effect to the Brazil-nut effect when the particle coefficient of friction increased after a long shaking time. Kondic et al. [17] demonstrated the occurrence of segregation by friction in a container with a small “hill” in the middle under horizontal shaking. However, Pohlman et al. [18] demonstrated that, although the repose angles of the rough and smooth particles may be different, radial segregation is still not found in the rotating drum. Zamankhan [19] indicated that friction coefficient of granular materials has significance on the formation of bubble in a vertical vibration bed. In addition, the filling bed height is also an essential parameter for investigating the dynamics of granular material in a vertical shaker. Hsiau et al. [20] indicated that the convection strength, convection size, and overall average granular temperature lead to a two-peak phenomenon with an increase of bed height. They also demonstrated that this two-peak phenomenon is because of the formation of a solid-like region in the granular bed. Bose and Rhodes [21] studied a regime in which the intruder was less dense than the bed-immersed granular materials, determining that bed height plays a significant role in the rise dynamics of the intruder. The vertical shaker is widely used in industries for drying, mixing, and segregating granular materials. Previous studies have shown that internal friction and bed height have a significant influence on dynamic properties and the Brazil-nut mechanism. However, the combined effects of internal friction and bed height on the Brazil-nut problem have not been examined simultaneously. In this study, we experimentally studied the combined effects of internal friction and filling bed height on the Brazil-nut problem in a quasi-2D vertical shaker. Imaging technology and a particle tracking method were employed to measure
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the rise dynamics of intruders in a vertical vibrating granular bed. The penetration length and friction drag force were also determined and are discussed in this paper. 2. Experimental setup Fig. 1 shows a schematic drawing of the experimental apparatus. An electromagnetic vibration system (Techron VTS-100) was employed as the vertical shaker driven using sinusoidal signals produced by a function generator (Meter Inc. DDS FG-503) through a power amplifier (Techron Mode 5530). The vibration frequency f and vibration acceleration a were measured using an accelerometer (Dytran 3136A) fixed to the shaker and connected to an oscilloscope (Tektronix TDS 210). The radian frequency ω and the amplitude A of the vibration could be calculated by ω = 2πf and A = a / ω2, respectively. The dimensionless vibration acceleration Γ is defined as Γ = a / g, where g is the gravitational acceleration. The bed container was equipped with glass plates at the front and back walls and plexiglass at the side and the bottom wall. The height, width, and depth of the inside of the container were 150.0 mm, 60.0 mm, and 4.0 mm, respectively. To record the intruder dynamic motion, the quasi 2D vibration granular bed was used to study the brazil-nut problem in this study. For the immersed granular material, we used smooth mono-sized glass beads. Each bead had a diameter d of 2 mm and a density ρp of 2.476 g/cm3. Three surface roughness cylindrical intruders with a diameter of 6.0 mm and a height of 3.0 mm were used in this study. The intruders were composed of stainless steel with the same density of 7.5 g/cm3. The curved surfaces of two intruders were coated with glue and covered with silica sand. Silica sand with size ranges of 425–500 μm and 212–250 μm (Fig. 2) was used to produce two grades of surface roughness. The flat surfaces of the intruders that face the front and back walls of the container did not have silica sand glued onto them; therefore, only the curved surfaces of the intruders interacted with the immersed glass beads. A commercial Jenike shearing tester was used to measure the friction of the intruders to quantify the surface roughness of the intruders. Because we could not use the prepared intruders to obtain friction coefficients and three stainless steel plates were used to replicate the surface roughness of the intruders. One was left smooth, whereas the other two had silica sand with sizes ranging from 425 to 500 μm and 212 to 250 μm glued onto them. These plates
were then used with the commercial Jenike shearing tester to measure the friction coefficient between the plates and the 2.0 mm glass beads. The friction coefficient for bead–wall interaction can be determined likewise. Using this method, we determined that the friction coefficients between the 2.0 mm glass beads and the walls were 0.143 (smooth), 0.405 (glued with 212–250 μm silica sand), and 0.523 (glued with 425–500 μm silica sand). Each case was repeated at least three times to calculate the average friction coefficient values. The value for the bead-wall friction coefficient can also be regarded as the same as that for the intruder-bead coefficient in the bed container. However, this value does not describe the true physical interaction between intruders and beads in the experiments. In 2D granular beds, rougher side-walls induce stronger convection cells [19, 21]. Emery paper (KA961 P60) was glued to the sidewalls to generate sufficient shear in the flow field. The intruders were placed at the center line 9 mm from the bottom of the container in all experimental runs (see Fig. 1). A high-speed charge-coupled device (CCD) camera (IDT X-3 Plus with a grabbing speed of 100 FPS) was employed to record the front view of the intruder motion. The CCD camera was set up in front of the vibration container, and the relative position of the camera and the vibration container is shown in Fig. 1. Using a particle tracking method accompanied by an image processing system, the positions and velocities of the intruder at different times could be measured [20, 22, 23–26]. The path of the intruder to the bed surface is shown in Fig. 3. A series of experiments was conducted with a vertical shaker using different friction coefficients of intruders and filling bed heights to investigate the Brazil-nut problem. The vibration was controlled at Γ = 3 and f = 35 Hz. The detailed experimental parameters are listed in Table 1. Each case was repeated at least three times to calculate the average rise-time values. 3. Result and discussions Fig. 4 shows the position of the intruder plotted as a function of time with different bed height hf and intruder-friction coefficient μp. In all cases, it shows that the intruder rises upward from the bottom to the free surface of the bed because of the Brazil-nut effect. The rise time was less with each reduction of the intruder's friction coefficient at each bed height. The motion of the rougher intruder was mitigated because of the larger drag force existing between the intruder and the
Fig. 1. The schematic drawing of the experimental apparatus.
C.-C. Liao et al. / Powder Technology 253 (2014) 561–567
Glued silica sands 425–500 μm
Glued silica sands 212–250 μm
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Smooth
Fig. 2. The images of friction coefficients of intruders.
immersed glass beads during ascent. Thus, the strength of the Brazil-nut effect was also reduced, resulting in a longer rise time by using the greater intruder-friction coefficient. By contrast, the smooth intruder had more kinetic energy because the interaction between the intruder and the immersed glass beads caused less energy dissipation, which allowed the intruder to overcome the resistance easily and rise faster. It was also found that the three intruders showed a two-stage rise process at the highest bed height, hf = 106 mm. The intruder rose gradually in the lower part of the bed, where the Brazil-nut effect is dominated by inertia mechanism [27, 28]; it then rose dramatically as the intruder entered the convection cell in the upper part of the bed, where the convection mechanism becomes the dominant parameter affecting intruder dynamics [2, 27, 29]. Granular compaction was stronger in the lower part of the bed because of the effects of gravity, and the drag force and the resistance between the intruder and the immersed glass beads were also larger at the relatively higher bed height. Hence, the intruder had less mobility and could not easily move upward when a larger bed height was applied to the system. The result is in agreement with the previous study [21]. In the convection region, the granular bed is more diluted and the influence of the friction between intruder and beads in the Brazil-nut effect is less critical. Driven by convection, the intruder rises easily and shows an approximately linear evolution in the upper part of the bed. It also shows that the slopes of curves with three different surface roughness intruders are similar in the upper part of granular bed. It means that the rise dynamics with different friction coefficients of intruders are similar in the upper part of granular bed. The convection strength is similar at the same vibration conditions. The result is similar to our previous study [2]. To quantify the rise time with different bed heights, the rise time was measured from the initial position to 40 mm of the bed height [2]. Fig. 5 shows the rise time plotted as a function of intruder-friction
coefficients with four different filling bed heights. The rise time was enhanced when the intruder-friction coefficient was increased. The intruder's kinetic energy dissipated because of the friction drag force resulting from the frictional interaction between the intruder and the immersed glass beads. In this study, the friction coefficient of immersed glass beads remained constant; therefore, the difference of the drag force between the intruder and the immersed glass beads was dominated by the friction coefficient of the intruder. Thus, the smoother intruder had a smaller friction coefficient and caused less energy dissipation because of the smaller drag force, providing this intruder with stronger mobility that resulted in a faster ascent that led to the smaller rise time, as shown in Fig. 5. It also indicated that the rise time is larger as the greater bed height is applied to the system. A higher bed height enhanced the frictional effect because of the relative weak particle motions and the less fluidization in the lower part of bed, whereas a lower bed height reduced it. Hence, the fluidization of the granular bed was fully developed with the smaller bed height, and the interactive collisions between particles might predominate over the particle motions based on our previous study [20]. The frictional interaction between the intruder and the immersed glass beads was not detrimental, leading to a smaller rise time with the smaller bed height, as shown in Fig. 5. From our previous studies [2, 20], a solid-like region could be formed in the lower part of bed, thus causing a stronger frictional effect as the larger bed height was applied. Hence, the difference of rise times with different intruder-friction coefficients is significant with the largest bed height. Fig. 6 shows the rise velocity plotted as a function of intruder-friction coefficients with different bed heights. As shown in Fig. 6, the rise velocity of the intruders decreases with the increase of the intruder-friction coefficients. In the preceding discussion, the smoother intruder caused the weaker drag force and the lessened energy dissipation during the
Fig. 3. Snapshots of the path of the smooth intruder rising to the bed surface (bed height = 106 mm) at Γ = 3, f = 35 Hz.
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250
Table 1 Parameters used in the current experiments. Γ
35
3
Bed height (mm)
μp μp μp μp μp μp μp μp μp μp μp μp
46
66
86
106
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
Friction coefficients of intruders = = = = = = = = = = = =
200
0.523 0.405 0.143 0.523 0.405 0.143 0.523 0.405 0.143 0.523 0.405 0.143
Rise time (sec)
Vibration frequency (Hz)
150
100
50
process of rising. Thus, the Brazil-nut effect was strengthened, which led to an increased rise velocity, indicating that the rise velocity is higher when the bed height is lower, but decreases when the bed height is higher at the same intruder-friction coefficient. The granular bed is more fluidized with the smaller bed height, allowing the intruder to rise up easily, resulting in the higher rise velocity. The result is similar to the previous study [21]. When the bed height is increased, the granular bed becomes denser in the lower part of the bed, causing greater intruder-bead resistance as friction becomes more significant and explaining why the rise velocity is lower as the larger bed height is applied to the system, as shown in Fig. 6. Nahmad-Molinari et al. [28] defined the penetration length Pl, noting that the intruder penetrates the granular bed by inertia for a small distance in each cycle. They also indicated that the kinetic energy of the intruder is dissipated by friction as it penetrates the granular bed on each cycle. Fig. 7 shows the position of the intruders plotted as a function of vibrating cycle (tω / 2π) at hf = 106 mm. Only the lower part of the curves is linearly fitted to avoid the convection effect in the upper bed. Based on the lines on the chart, we could determine the slope that represents the penetration length at each vibration cycle [2, 28]. According to the model based on the simple energy balance per cycle [28], 1 2 mvto ¼ βP l 2
ð1Þ
0 0.1
0.2
0.3
μp
0.4
is simply the values of z˙ðt Þ when zðt Þ ¼ −g, where z(t) = A sin(ωt). Hence, h i 2 2 2 2 1=2 vto ¼ A ω −g =ω :
ð2Þ
Considering Eq. (1), the kinetic energy of the intruder is mainly dissipated by the drag existing between the intruder and the immersed glass beads in each vibration cycle. It is noted that the simple model could be applied in the dense particle packing where the friction effect has a significant influence on particle motions. In this study, we used this simple model to determine the penetration length and drag force only considering the intruder dynamics in the lower part of granular bed where the particle motions is weak and less fluidization, causing the serious friction effect between particles. The penetration length is plotted as a function of intruder-friction coefficients with different bed 8
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
7 120
Rise velocity (mm/sec)
6
Height (mm)
80
60
40
hf = 106 mm hf = 86 mm hf = 66 mm hf = 46 mm μp = 0.523 μp = 0.405 μp = 0.143
20
50
100
150
200
250
300
5
4
3
2
1
0
0
0.6
Fig. 5. The rise time plotted as a function of friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
where m is the mass of intruder, vto is the take-off velocity of intruder as the system reaches a negative acceleration a = − g, β the drag force between the beads and the intruder, and Pl the penetration length. vto
100
0.5
350
Time (sec) Fig. 4. The position of the intruder plotted as a function of vibrating time with different bed heights at Γ = 3, f = 35 Hz.
0 0.1
0.2
0.3
μp
0.4
0.5
0.6
Fig. 6. The rise velocity plotted as a function of friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
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30
25
Height (mm)
20
15
10 μp = 0.523 μp = 0.405 μp = 0.143
5
0 0
1000
2000
3000
4000
5000
6000
7000
Cycle Fig. 7. The position of intruder plotted as a function of vibrating cycle (tw / 2π) at hf = 106 mm and Γ = 3, f = 35 Hz. The lines indicate the linear fits to the data by using the least-squares method.
heights, as shown in Fig. 8. Greater intruder-friction coefficients result in greater drag. Consequently, the substantial frictional effect between beads and intruder could occur during the rise. Hence, the smoother intruder has better mobility and can easily move upward, resulting in the longer penetration length [2]. More kinetic energy was dissipated because of the serious frictional interaction between the intruder and the immersed glass beads with the larger intruder-friction coefficient. Consequently, the penetration length of intruders was shorter when the intruder-friction coefficient was larger in each vibration cycle. It can also be seen that the penetration length was shortest at the greatest bed height and that it increased as the bed height in each intruderfriction coefficient was reduced. At the largest bed height, the solidlike region is formed in the lower part of bed and the sliding and rolling contact in a quasi-static state is the dominant interaction mechanism between particles [2, 20]. Hence, the intruders' motion was hindered.
Additionally, the granular bed was more fluidized with the decreasing bed height. The resistance between the intruder and the beads also mitigated because of the better fluidization in the granular bed. Therefore, the Brazil-nut effect is stronger and has a longer penetration length with the smaller bed height. According to Eq. (1), the penetration length is inversely proportional to drag force. Rougher intruders have greater β, leading to the higher kinetic energy dissipation, resulting in the shorter penetration lengths. To test our physical argument, the drag force β was also calculated in this study according to Eq. (1). The drag force was plotted as a function of intruder-friction coefficients with different bed heights, as shown in Fig. 9. It shows that the drag force β increases with the increase of the intruder-friction coefficient. The result is in agreement with our preceding physical arguments. The larger intruder-friction coefficient had a greater resistance and the stronger frictional interaction between the intruder and the immersed glass beads. Hence, the drag force β was larger when the rougher intruders were used. It also shows that the drag force was reduced with the decrease of the bed height in the specific intruder-friction coefficient. The strength of particle motions and interactive collisions was larger and caused full fluidization in the granular bed at the smaller bed height. Hence, the frictional effect between the intruder and the beads was less pronounced with the full fluidization conditions and resulted in the smaller drag force with the smaller bed height. It could also demonstrate why the drag force was small and increased slightly even though the intruder-friction coefficient was increased from μp = 0.143 to μp = 0.523 with the lowest bed height, as shown in Fig. 9. Fig. 10 shows the intruder rise times plotted as a function of bed height with three intruder-friction coefficients: μp = 0.523, μp = 0.405, and μp = 0.143. It shows that the rise time became greater with the increasing bed height in each intruder-friction coefficient. The result is in agreement with a previous study [21]. From our previous studies [2, 20], the solid-like region could be thicker and cause the stronger granular compaction and the weaker particle motions and smaller granular temperature in the lower part of the bed because of the effect of gravity in the larger bed height. Hence, the rise time was longer as the larger bed height was applied to the granular system due to the serious friction effect between the intruder and the immersed glass beads in the lower part of granular bed. The result is also in agreement with previous study that the slow initial rising period of intruder increases as the bed height is increased due to the stronger
0.15
3
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
2.5
0.1
2
β(N)
Penetration length, Pl (mm)
565
0.05
1.5
1
0.5
0 0.1
0 0.2
0.3
μp
0.4
0.5
0.6
Fig. 8. The penetration length plotted as a function of friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
0.1
0.2
0.3
μp
0.4
0.5
0.6
Fig. 9. The β plotted against the friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
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250 μp = 0.523 μp = 0.405 μp = 0.143
200
Rise time (sec)
Nomenclature A vibration amplitude a vibration acceleration d bead diameter f vibration frequency g gravitational acceleration hf bed height m mass of intruder Pl penetration length vto take-off velocity of intruder Γ dimensionless vibration acceleration Β drag force μp intruder-friction coefficient ρp bead density ω radian frequency
150
100
50
0
Acknowledgment 40
50
60
70
80
90
100
110
Bed height (mm) Fig. 10. The rise time plotted against the bed height with different friction coefficients of intruder at Γ = 3, f = 35 Hz.
resistance between intruder and the immersed glass beads [21]. Fig. 10 also indicates that the greatest intruder-friction coefficient causes the longest rise time in the specific bed height. This is also consistent with the physical arguments, that the kinetic energy of the intruder is dissipated because of the greater drag force resulting from the rougher intruder. Additionally, we also found that the frictional effect had a significant influence on the rise time with the larger bed height. The influence of the frictional effect was mitigated as the bed height is reduced. The granular bed was easy to fluidize and the solid-like region could not be formed, therefore the spaces between particles were also larger at the smaller bed height. Consequently, the frictional effect was less important on the intruders' rise dynamics and the rise time was similar for the three intruder-friction coefficients with the smallest bed height, as shown in Fig. 10. At the higher filling bed height, the solid-like region could be generated in the lower part of the granular bed [20]. The particle motion was weak and the rolling and sliding contact was the main interactive types between particles in the solid-like region (quasi-static state). In this case, the drag force had a significant influence on the rise dynamics of intruders. Consequently, the variation of rise time with three intruder-friction coefficients was enhanced as the bed height increased, as shown in Fig. 10.
4. Conclusion We performed a series of experiments to investigate the combined effects of intruder-friction coefficients and filling bed height on the Brazil-nut problem in a quasi-2D vertical shaker. The motion of the intruder was recorded by a high-speed camera and the rise dynamics were determined and then were discussed in this paper. The results demonstrate that the “Brazil-nut effect” could be mitigated because of the larger intruder-friction coefficient and the higher filling bed height. The results show that the rising time increased with the increase of the intruder-friction coefficient and that it decreased with the reduction of filling bed height. The penetration length was reduced and drag force was enhanced with the increase of the intruder-friction coefficient and the filling bed height. In addition, the influence of the frictional effect on the Brazil-nut mechanism was not significant with the smaller filling bed height. However, the frictional effect played a crucial role in the rise dynamics of intruders as the larger bed height was applied to the granular system.
The authors would like to acknowledge the support from the National Science Council of the ROC for this work through Grant NSC 100-2221-E-008-078-MY3.
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Combined effects of internal friction and bed height on the Brazil-nut problem in a shaker Chun-Chung Liao, Shu-San Hsiau ⁎, Chi-Sou Wu Department of Mechanical Engineering, National Central University, No. 300, Jhongda Road, Jhongli 32001, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 26 August 2013 Received in revised form 2 December 2013 Accepted 14 December 2013 Available online 21 December 2013 Keywords: Penetration length Friction drag force Granular material Brazil-nut problem Vibration bed
a b s t r a c t We experimentally studied the influence of an intruder's friction coefficient and bed-filling height on the Brazil-nut effect in a quasi-2D vertical vibration granular bed. The motion of intruder was successfully measured using a high-speed camera and the rising time of intruder was determined by using a particle tracking method with the help of image processing technology. The results show that an intruder's friction coefficient and filling bed height play significant roles in the rise dynamics. The results also show that the rise time increases when the intruder's friction coefficient increases, which is reduced when the filling bed height decreases. Penetration length and friction drag force were also determined in this study. The penetration length was reduced and the friction drag force was enhanced with the increase of an intruder's friction coefficient and bed height. Additionally, the variation between the rise times of the smooth and rough intruders was not significant with the lower bed-filling heights. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Granular materials may flow like a liquid or a gas, or behave like a solid. These three motion states may occur simultaneously or exist individually in granular systems [1]. Granular materials are ubiquitous in nature and exhibit a wide range of nontrivial rheological behavior. The rheological behavior of granular materials is complex, but an understanding of granular materials is imperative in many industries, such as the pharmaceutical industry, food storage and transportation, polymer manufacturing, cement manufacturing, and metallurgy. Furthermore, an understanding of granular materials can also be applied to environmental problems such as debris flows and landslides. The segregation phenomenon of granular materials is a critical issue in industrial processes, and this fascinating phenomenon also occurs in vibrated granular beds. The so-called “Brazil-nut problem,” which involves an exceptionally large particle immersed with smaller granular materials in a container that vibrated using a shaker, has been widely studied in recent decades [2–16]. It is well-known that segregation can occur in binary mixtures that have components of different sizes, densities, surface roughnesses, and restitution coefficients. Understanding the segregation mechanisms of granular material is a key scientific and technological challenge to both engineers and scientists. Granular segregation induced by internal friction is a central issue in many industries. Srebro and Levine [14] observed segregation at compactivities in binary mixtures of grains exhibiting differing frictional properties. Plantard et al. [13] observed the occurrence of frictioninduced segregation in a granular slurry shear system, finding that ⁎ Corresponding author. Tel.: +886 3 426 7341; fax: +886 3 425 4501. E-mail address: [email protected] (S.-S. Hsiau). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.12.031
rough circular particles behave similarly to smooth but larger circular particles. Ulrich et al. [16] experimentally demonstrated the transition from the reverse Brazil-nut effect to the Brazil-nut effect when the particle coefficient of friction increased after a long shaking time. Kondic et al. [17] demonstrated the occurrence of segregation by friction in a container with a small “hill” in the middle under horizontal shaking. However, Pohlman et al. [18] demonstrated that, although the repose angles of the rough and smooth particles may be different, radial segregation is still not found in the rotating drum. Zamankhan [19] indicated that friction coefficient of granular materials has significance on the formation of bubble in a vertical vibration bed. In addition, the filling bed height is also an essential parameter for investigating the dynamics of granular material in a vertical shaker. Hsiau et al. [20] indicated that the convection strength, convection size, and overall average granular temperature lead to a two-peak phenomenon with an increase of bed height. They also demonstrated that this two-peak phenomenon is because of the formation of a solid-like region in the granular bed. Bose and Rhodes [21] studied a regime in which the intruder was less dense than the bed-immersed granular materials, determining that bed height plays a significant role in the rise dynamics of the intruder. The vertical shaker is widely used in industries for drying, mixing, and segregating granular materials. Previous studies have shown that internal friction and bed height have a significant influence on dynamic properties and the Brazil-nut mechanism. However, the combined effects of internal friction and bed height on the Brazil-nut problem have not been examined simultaneously. In this study, we experimentally studied the combined effects of internal friction and filling bed height on the Brazil-nut problem in a quasi-2D vertical shaker. Imaging technology and a particle tracking method were employed to measure
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the rise dynamics of intruders in a vertical vibrating granular bed. The penetration length and friction drag force were also determined and are discussed in this paper. 2. Experimental setup Fig. 1 shows a schematic drawing of the experimental apparatus. An electromagnetic vibration system (Techron VTS-100) was employed as the vertical shaker driven using sinusoidal signals produced by a function generator (Meter Inc. DDS FG-503) through a power amplifier (Techron Mode 5530). The vibration frequency f and vibration acceleration a were measured using an accelerometer (Dytran 3136A) fixed to the shaker and connected to an oscilloscope (Tektronix TDS 210). The radian frequency ω and the amplitude A of the vibration could be calculated by ω = 2πf and A = a / ω2, respectively. The dimensionless vibration acceleration Γ is defined as Γ = a / g, where g is the gravitational acceleration. The bed container was equipped with glass plates at the front and back walls and plexiglass at the side and the bottom wall. The height, width, and depth of the inside of the container were 150.0 mm, 60.0 mm, and 4.0 mm, respectively. To record the intruder dynamic motion, the quasi 2D vibration granular bed was used to study the brazil-nut problem in this study. For the immersed granular material, we used smooth mono-sized glass beads. Each bead had a diameter d of 2 mm and a density ρp of 2.476 g/cm3. Three surface roughness cylindrical intruders with a diameter of 6.0 mm and a height of 3.0 mm were used in this study. The intruders were composed of stainless steel with the same density of 7.5 g/cm3. The curved surfaces of two intruders were coated with glue and covered with silica sand. Silica sand with size ranges of 425–500 μm and 212–250 μm (Fig. 2) was used to produce two grades of surface roughness. The flat surfaces of the intruders that face the front and back walls of the container did not have silica sand glued onto them; therefore, only the curved surfaces of the intruders interacted with the immersed glass beads. A commercial Jenike shearing tester was used to measure the friction of the intruders to quantify the surface roughness of the intruders. Because we could not use the prepared intruders to obtain friction coefficients and three stainless steel plates were used to replicate the surface roughness of the intruders. One was left smooth, whereas the other two had silica sand with sizes ranging from 425 to 500 μm and 212 to 250 μm glued onto them. These plates
were then used with the commercial Jenike shearing tester to measure the friction coefficient between the plates and the 2.0 mm glass beads. The friction coefficient for bead–wall interaction can be determined likewise. Using this method, we determined that the friction coefficients between the 2.0 mm glass beads and the walls were 0.143 (smooth), 0.405 (glued with 212–250 μm silica sand), and 0.523 (glued with 425–500 μm silica sand). Each case was repeated at least three times to calculate the average friction coefficient values. The value for the bead-wall friction coefficient can also be regarded as the same as that for the intruder-bead coefficient in the bed container. However, this value does not describe the true physical interaction between intruders and beads in the experiments. In 2D granular beds, rougher side-walls induce stronger convection cells [19, 21]. Emery paper (KA961 P60) was glued to the sidewalls to generate sufficient shear in the flow field. The intruders were placed at the center line 9 mm from the bottom of the container in all experimental runs (see Fig. 1). A high-speed charge-coupled device (CCD) camera (IDT X-3 Plus with a grabbing speed of 100 FPS) was employed to record the front view of the intruder motion. The CCD camera was set up in front of the vibration container, and the relative position of the camera and the vibration container is shown in Fig. 1. Using a particle tracking method accompanied by an image processing system, the positions and velocities of the intruder at different times could be measured [20, 22, 23–26]. The path of the intruder to the bed surface is shown in Fig. 3. A series of experiments was conducted with a vertical shaker using different friction coefficients of intruders and filling bed heights to investigate the Brazil-nut problem. The vibration was controlled at Γ = 3 and f = 35 Hz. The detailed experimental parameters are listed in Table 1. Each case was repeated at least three times to calculate the average rise-time values. 3. Result and discussions Fig. 4 shows the position of the intruder plotted as a function of time with different bed height hf and intruder-friction coefficient μp. In all cases, it shows that the intruder rises upward from the bottom to the free surface of the bed because of the Brazil-nut effect. The rise time was less with each reduction of the intruder's friction coefficient at each bed height. The motion of the rougher intruder was mitigated because of the larger drag force existing between the intruder and the
Fig. 1. The schematic drawing of the experimental apparatus.
C.-C. Liao et al. / Powder Technology 253 (2014) 561–567
Glued silica sands 425–500 μm
Glued silica sands 212–250 μm
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Smooth
Fig. 2. The images of friction coefficients of intruders.
immersed glass beads during ascent. Thus, the strength of the Brazil-nut effect was also reduced, resulting in a longer rise time by using the greater intruder-friction coefficient. By contrast, the smooth intruder had more kinetic energy because the interaction between the intruder and the immersed glass beads caused less energy dissipation, which allowed the intruder to overcome the resistance easily and rise faster. It was also found that the three intruders showed a two-stage rise process at the highest bed height, hf = 106 mm. The intruder rose gradually in the lower part of the bed, where the Brazil-nut effect is dominated by inertia mechanism [27, 28]; it then rose dramatically as the intruder entered the convection cell in the upper part of the bed, where the convection mechanism becomes the dominant parameter affecting intruder dynamics [2, 27, 29]. Granular compaction was stronger in the lower part of the bed because of the effects of gravity, and the drag force and the resistance between the intruder and the immersed glass beads were also larger at the relatively higher bed height. Hence, the intruder had less mobility and could not easily move upward when a larger bed height was applied to the system. The result is in agreement with the previous study [21]. In the convection region, the granular bed is more diluted and the influence of the friction between intruder and beads in the Brazil-nut effect is less critical. Driven by convection, the intruder rises easily and shows an approximately linear evolution in the upper part of the bed. It also shows that the slopes of curves with three different surface roughness intruders are similar in the upper part of granular bed. It means that the rise dynamics with different friction coefficients of intruders are similar in the upper part of granular bed. The convection strength is similar at the same vibration conditions. The result is similar to our previous study [2]. To quantify the rise time with different bed heights, the rise time was measured from the initial position to 40 mm of the bed height [2]. Fig. 5 shows the rise time plotted as a function of intruder-friction
coefficients with four different filling bed heights. The rise time was enhanced when the intruder-friction coefficient was increased. The intruder's kinetic energy dissipated because of the friction drag force resulting from the frictional interaction between the intruder and the immersed glass beads. In this study, the friction coefficient of immersed glass beads remained constant; therefore, the difference of the drag force between the intruder and the immersed glass beads was dominated by the friction coefficient of the intruder. Thus, the smoother intruder had a smaller friction coefficient and caused less energy dissipation because of the smaller drag force, providing this intruder with stronger mobility that resulted in a faster ascent that led to the smaller rise time, as shown in Fig. 5. It also indicated that the rise time is larger as the greater bed height is applied to the system. A higher bed height enhanced the frictional effect because of the relative weak particle motions and the less fluidization in the lower part of bed, whereas a lower bed height reduced it. Hence, the fluidization of the granular bed was fully developed with the smaller bed height, and the interactive collisions between particles might predominate over the particle motions based on our previous study [20]. The frictional interaction between the intruder and the immersed glass beads was not detrimental, leading to a smaller rise time with the smaller bed height, as shown in Fig. 5. From our previous studies [2, 20], a solid-like region could be formed in the lower part of bed, thus causing a stronger frictional effect as the larger bed height was applied. Hence, the difference of rise times with different intruder-friction coefficients is significant with the largest bed height. Fig. 6 shows the rise velocity plotted as a function of intruder-friction coefficients with different bed heights. As shown in Fig. 6, the rise velocity of the intruders decreases with the increase of the intruder-friction coefficients. In the preceding discussion, the smoother intruder caused the weaker drag force and the lessened energy dissipation during the
Fig. 3. Snapshots of the path of the smooth intruder rising to the bed surface (bed height = 106 mm) at Γ = 3, f = 35 Hz.
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250
Table 1 Parameters used in the current experiments. Γ
35
3
Bed height (mm)
μp μp μp μp μp μp μp μp μp μp μp μp
46
66
86
106
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
Friction coefficients of intruders = = = = = = = = = = = =
200
0.523 0.405 0.143 0.523 0.405 0.143 0.523 0.405 0.143 0.523 0.405 0.143
Rise time (sec)
Vibration frequency (Hz)
150
100
50
process of rising. Thus, the Brazil-nut effect was strengthened, which led to an increased rise velocity, indicating that the rise velocity is higher when the bed height is lower, but decreases when the bed height is higher at the same intruder-friction coefficient. The granular bed is more fluidized with the smaller bed height, allowing the intruder to rise up easily, resulting in the higher rise velocity. The result is similar to the previous study [21]. When the bed height is increased, the granular bed becomes denser in the lower part of the bed, causing greater intruder-bead resistance as friction becomes more significant and explaining why the rise velocity is lower as the larger bed height is applied to the system, as shown in Fig. 6. Nahmad-Molinari et al. [28] defined the penetration length Pl, noting that the intruder penetrates the granular bed by inertia for a small distance in each cycle. They also indicated that the kinetic energy of the intruder is dissipated by friction as it penetrates the granular bed on each cycle. Fig. 7 shows the position of the intruders plotted as a function of vibrating cycle (tω / 2π) at hf = 106 mm. Only the lower part of the curves is linearly fitted to avoid the convection effect in the upper bed. Based on the lines on the chart, we could determine the slope that represents the penetration length at each vibration cycle [2, 28]. According to the model based on the simple energy balance per cycle [28], 1 2 mvto ¼ βP l 2
ð1Þ
0 0.1
0.2
0.3
μp
0.4
is simply the values of z˙ðt Þ when zðt Þ ¼ −g, where z(t) = A sin(ωt). Hence, h i 2 2 2 2 1=2 vto ¼ A ω −g =ω :
ð2Þ
Considering Eq. (1), the kinetic energy of the intruder is mainly dissipated by the drag existing between the intruder and the immersed glass beads in each vibration cycle. It is noted that the simple model could be applied in the dense particle packing where the friction effect has a significant influence on particle motions. In this study, we used this simple model to determine the penetration length and drag force only considering the intruder dynamics in the lower part of granular bed where the particle motions is weak and less fluidization, causing the serious friction effect between particles. The penetration length is plotted as a function of intruder-friction coefficients with different bed 8
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
7 120
Rise velocity (mm/sec)
6
Height (mm)
80
60
40
hf = 106 mm hf = 86 mm hf = 66 mm hf = 46 mm μp = 0.523 μp = 0.405 μp = 0.143
20
50
100
150
200
250
300
5
4
3
2
1
0
0
0.6
Fig. 5. The rise time plotted as a function of friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
where m is the mass of intruder, vto is the take-off velocity of intruder as the system reaches a negative acceleration a = − g, β the drag force between the beads and the intruder, and Pl the penetration length. vto
100
0.5
350
Time (sec) Fig. 4. The position of the intruder plotted as a function of vibrating time with different bed heights at Γ = 3, f = 35 Hz.
0 0.1
0.2
0.3
μp
0.4
0.5
0.6
Fig. 6. The rise velocity plotted as a function of friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
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30
25
Height (mm)
20
15
10 μp = 0.523 μp = 0.405 μp = 0.143
5
0 0
1000
2000
3000
4000
5000
6000
7000
Cycle Fig. 7. The position of intruder plotted as a function of vibrating cycle (tw / 2π) at hf = 106 mm and Γ = 3, f = 35 Hz. The lines indicate the linear fits to the data by using the least-squares method.
heights, as shown in Fig. 8. Greater intruder-friction coefficients result in greater drag. Consequently, the substantial frictional effect between beads and intruder could occur during the rise. Hence, the smoother intruder has better mobility and can easily move upward, resulting in the longer penetration length [2]. More kinetic energy was dissipated because of the serious frictional interaction between the intruder and the immersed glass beads with the larger intruder-friction coefficient. Consequently, the penetration length of intruders was shorter when the intruder-friction coefficient was larger in each vibration cycle. It can also be seen that the penetration length was shortest at the greatest bed height and that it increased as the bed height in each intruderfriction coefficient was reduced. At the largest bed height, the solidlike region is formed in the lower part of bed and the sliding and rolling contact in a quasi-static state is the dominant interaction mechanism between particles [2, 20]. Hence, the intruders' motion was hindered.
Additionally, the granular bed was more fluidized with the decreasing bed height. The resistance between the intruder and the beads also mitigated because of the better fluidization in the granular bed. Therefore, the Brazil-nut effect is stronger and has a longer penetration length with the smaller bed height. According to Eq. (1), the penetration length is inversely proportional to drag force. Rougher intruders have greater β, leading to the higher kinetic energy dissipation, resulting in the shorter penetration lengths. To test our physical argument, the drag force β was also calculated in this study according to Eq. (1). The drag force was plotted as a function of intruder-friction coefficients with different bed heights, as shown in Fig. 9. It shows that the drag force β increases with the increase of the intruder-friction coefficient. The result is in agreement with our preceding physical arguments. The larger intruder-friction coefficient had a greater resistance and the stronger frictional interaction between the intruder and the immersed glass beads. Hence, the drag force β was larger when the rougher intruders were used. It also shows that the drag force was reduced with the decrease of the bed height in the specific intruder-friction coefficient. The strength of particle motions and interactive collisions was larger and caused full fluidization in the granular bed at the smaller bed height. Hence, the frictional effect between the intruder and the beads was less pronounced with the full fluidization conditions and resulted in the smaller drag force with the smaller bed height. It could also demonstrate why the drag force was small and increased slightly even though the intruder-friction coefficient was increased from μp = 0.143 to μp = 0.523 with the lowest bed height, as shown in Fig. 9. Fig. 10 shows the intruder rise times plotted as a function of bed height with three intruder-friction coefficients: μp = 0.523, μp = 0.405, and μp = 0.143. It shows that the rise time became greater with the increasing bed height in each intruder-friction coefficient. The result is in agreement with a previous study [21]. From our previous studies [2, 20], the solid-like region could be thicker and cause the stronger granular compaction and the weaker particle motions and smaller granular temperature in the lower part of the bed because of the effect of gravity in the larger bed height. Hence, the rise time was longer as the larger bed height was applied to the granular system due to the serious friction effect between the intruder and the immersed glass beads in the lower part of granular bed. The result is also in agreement with previous study that the slow initial rising period of intruder increases as the bed height is increased due to the stronger
0.15
3
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
hf = 46 mm hf = 66 mm hf = 86 mm hf = 106 mm
2.5
0.1
2
β(N)
Penetration length, Pl (mm)
565
0.05
1.5
1
0.5
0 0.1
0 0.2
0.3
μp
0.4
0.5
0.6
Fig. 8. The penetration length plotted as a function of friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
0.1
0.2
0.3
μp
0.4
0.5
0.6
Fig. 9. The β plotted against the friction coefficient of intruder with different bed heights at Γ = 3, f = 35 Hz.
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250 μp = 0.523 μp = 0.405 μp = 0.143
200
Rise time (sec)
Nomenclature A vibration amplitude a vibration acceleration d bead diameter f vibration frequency g gravitational acceleration hf bed height m mass of intruder Pl penetration length vto take-off velocity of intruder Γ dimensionless vibration acceleration Β drag force μp intruder-friction coefficient ρp bead density ω radian frequency
150
100
50
0
Acknowledgment 40
50
60
70
80
90
100
110
Bed height (mm) Fig. 10. The rise time plotted against the bed height with different friction coefficients of intruder at Γ = 3, f = 35 Hz.
resistance between intruder and the immersed glass beads [21]. Fig. 10 also indicates that the greatest intruder-friction coefficient causes the longest rise time in the specific bed height. This is also consistent with the physical arguments, that the kinetic energy of the intruder is dissipated because of the greater drag force resulting from the rougher intruder. Additionally, we also found that the frictional effect had a significant influence on the rise time with the larger bed height. The influence of the frictional effect was mitigated as the bed height is reduced. The granular bed was easy to fluidize and the solid-like region could not be formed, therefore the spaces between particles were also larger at the smaller bed height. Consequently, the frictional effect was less important on the intruders' rise dynamics and the rise time was similar for the three intruder-friction coefficients with the smallest bed height, as shown in Fig. 10. At the higher filling bed height, the solid-like region could be generated in the lower part of the granular bed [20]. The particle motion was weak and the rolling and sliding contact was the main interactive types between particles in the solid-like region (quasi-static state). In this case, the drag force had a significant influence on the rise dynamics of intruders. Consequently, the variation of rise time with three intruder-friction coefficients was enhanced as the bed height increased, as shown in Fig. 10.
4. Conclusion We performed a series of experiments to investigate the combined effects of intruder-friction coefficients and filling bed height on the Brazil-nut problem in a quasi-2D vertical shaker. The motion of the intruder was recorded by a high-speed camera and the rise dynamics were determined and then were discussed in this paper. The results demonstrate that the “Brazil-nut effect” could be mitigated because of the larger intruder-friction coefficient and the higher filling bed height. The results show that the rising time increased with the increase of the intruder-friction coefficient and that it decreased with the reduction of filling bed height. The penetration length was reduced and drag force was enhanced with the increase of the intruder-friction coefficient and the filling bed height. In addition, the influence of the frictional effect on the Brazil-nut mechanism was not significant with the smaller filling bed height. However, the frictional effect played a crucial role in the rise dynamics of intruders as the larger bed height was applied to the granular system.
The authors would like to acknowledge the support from the National Science Council of the ROC for this work through Grant NSC 100-2221-E-008-078-MY3.
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