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Dynamic properties of immersed granular matter in different flow regimes in a rotating drum
Powder Technology 226 (2012) 99–106
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Dynamic properties of immersed granular matter in different flow regimes in a rotating drum S.H. Chou, S.S. Hsiau ⁎ Department of Mechanical Engineering, National Central University, Jhongli 32001, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 30 January 2012 Received in revised form 26 March 2012 Accepted 12 April 2012 Available online 21 April 2012 Keywords: Rotating drum Rotation speed Filling degree Liquid viscosity Flow regime
a b s t r a c t Using particle tracking velocimetry, we study the dynamic properties and flowing behavior of immersed granular matter in a rotating drum. In this study, the interstitial fluid is water or a water–glycerol mixture. The filling degree of the particles, the rotation speed, and the viscosity of the interstitial fluid are the three experimental control parameters. The results show that both the granular dynamic properties and flowing behavior are strongly affected by the operational parameters. At lower rotation speed or liquid viscosity, the distance between the centroid of all particles and the center of the tank is the same as the initial configuration, but the distance decreases rapidly when the liquid viscosity is above a critical value. The mean velocity, obtained by averaging the velocities of all particles will decrease with the increase of the liquid viscosity. When the liquid viscosity is above a critical value, the mean velocity will increase, and the granular flow behavior will transform into a suspension regime. Furthermore, the experimental results indicate that the liquid viscosity and the flow rate per unit width have a significant influence on the dynamic properties and flow behavior of the immersed granular matter. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Rotating drums are usually simply cylinders rotating about its central axis so as to drive particle motion. Devices based on this configuration are widely used for processing granular materials in the mineral, pharmaceutical, ceramic, cement and food industries, in which they are used to perform drying, heating [1,2], chemical reactions [3–5], mixing and segregation [6–8]. Although the devices are simple and can be operated relatively easily, granular dynamic behavior is more complicated. Granular flow behavior in a rotating drum can be identified by several flow regimes based on the particle motion. The particle flow behavior and mixing/segregation mechanisms may be different in each flow regime. Six identifiable flow regimes in a dry granular system may be used to describe the particle motion in a rotating drum depending on different operational conditions, including the rotational speed, wall friction coefficient, filling degree, and so on. The flow regimes include slipping, slumping, rolling, cascading, cataracting and centrifuging [9,10], as shown in Fig. 1. However in the past, most studies have been on dry granular systems, where the interstitial fluid is air. It has been found that the investigation of particle transport properties is about the influence of different particle materials, size of particles or tank, even the rotation speed of the rotating drum, while the fluid (air) effects are small enough to be neglected. Jain et al. [11] used particle tracking
⁎ 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 © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2012.04.024
velocimetry to investigate the velocity profile in the fluidized layer with different particle sizes. They found that the streamwise velocity profile is nearly linear near the top of the flowing layer, while close to the interface between the flowing layer and fixed bed the velocity logarithmically decreases to zero. Furthermore, near the surface of the flowing layer, the velocity fluctuations can be as large as one-third of the maximum streamwise velocity. Boating and Barr [12] used optical-fiber probes to measure the mean velocities with different particle materials in different sizes of rotating drum. The results show that the bulk viscosity is proportional to the shear viscosity and dilation and inversely proportional to particle size. Most computer simulation studies have also focused on dry granular systems. Lu and Hsiau [13] used the discrete element method (DEM) to investigate the mixing behavior of particles in a sheared granular flow. They found that the mixing layer thickness is greater, and the mixing growth rate is faster in the upper part of the test section. Yang et al. [14] used DEM to investigate the microdynamic variables, including the porosity, coordination number, collision velocity and collision frequency in the rotating drum. In the past few years, some researchers have begun to focus on investigating the effects of liquid viscosity on particle flowing behavior in a slurry granular system [15–17]. When the interstitial fluid is liquid, the fluid (liquid) effect cannot be neglected. Medved et al. [18] investigated the convection in a granular system completely immersed in fluid inside a vertical cylindrical container. In comparison with a dry system, they found that the wet system flows up to two orders of magnitude more slowly than the dry one dose. Fiedor and Ottion [19] studied the dynamics of axial segregation and coarsening of dry
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Slipping
Slumping
Cascading
Rolling
Cataracting
Centrifuging
Fig. 1. Forms of transverse motion of solids in a rotating drum.
granular materials and slurries in tubes. They indicated that the fraction of surface area of small-rich particles increases resulting from the increasing viscosity of interstitial fluid. Jain et al. [20] used particle tracking velocimetry to measure the velocity field in a slurry system with different Froude numbers, bead sizes, fluid densities and fluid viscosity. They found that the thickness of the flowing layer and angle of repose with liquid as the interstitial fluid are generally larger than in the dry system with similar conditions, but the shear rate is generally smaller. Finger and Stannarius [21] investigated the influence of the viscosity of the interstitial liquid in a horizontally rotating mixer. They found that the viscosity of interstitial liquid played a crucial role in pattern dynamics and the structure of the segregation patterns. They also indicated that the density effect of interstitial fluid dose not influence the formation and evolution of segregation patterns of the granular materials. Liao et al. [22] studied the effects of interstitial fluid viscosity on the rates of dynamic processes in a thin rotating drum. They indicated that the characteristic speed of a bead in the flowing layer decreases with the fluid viscosity, but the mixing rate of beads is found to increase with fluid viscosity. Chou et al. [23] investigated the phenomena of particle segregation and flowing behavior in a slurry rotating drum with liquids having different viscosities and different filling degrees. They indicated that the segregation index and angle of repose are shown to decrease with increased liquid viscosity. When the liquid viscosity is the same, the increase in the filling degree causes the segregation index to increase, while the net rate of mixing seems to decrease. We found that the fluid kinematic viscosity is a very important parameter determining the flow behavior if the particles are fully immersed in liquid, but many unknown physical mechanisms exist. Therefore, in this study, we use experimental methods to obtain the dynamic properties of immersed granular matter in a quasi-twodimensional slurry rotating drum, where the interstitial fluid is water or a glycerol–water mixture. Our intent is to determine the macrodynamic variables, including the particle mean velocity, average granular temperature, and flowing behavior through particle tracking velocimetry. The filling degree of particles, the rotational speed and the density and viscosity of the interstitial fluid are different. 2. Experimental procedure A schematic representation of the circular drum used in the quasitwo-dimensional experiments is shown in Fig. 2. The diameter of the drum is 0.2 m and the gap width W is 0.02 m. The rear surface of the drum was constructed of a black anodized aluminum plate to minimize electrostatic effects on the particles and optical noise effects in the digital images. The front faceplate was made of clear acrylic to permit optical access. A small hole in the side of the tank permitted liquid to be injected from a hopper. A stepper motor and micro series driver combination was used to rotate the drum at a speed of 0.1047 rad/s, 0.2093 rad/s, 0.3140 rad/s, and 0.4187 rad/s, corresponding to the Froude number, Fr = Rω 2/g of 1.67 × 10 − 4, 6.71 × 10 − 4, 1.51 × 10 − 3, and 2.68 × 10 − 3, where ω is the angular velocity of the drum (=2π/T, T = rotation period); R is the radius of the drum; and g is the acceleration of gravity. The dimensionless axial
thickness of the drum, defined as the ratio of the drum's axial length and the big particle diameter, was set to 5 in this study. In each experiment, mono-sized glass beads were used as the granular material. The diameter of the glass beads was 4 mm with a standard deviation of 0.09 mm and their density ρ was 2.5 g/cm 3. Details of the experimental conditions are provided in Table 1. In all of the slurry flow experiments, the drum was completely filled (no air content) with a water–glycerin mixture before it is sealed. Some experiments were carried out with pure water as the interstitial fluid. In addition, several experiments were performed with mixtures of water and glycerin in a range of different viscosities. We use water–glycerin mixtures with different glycerin weight fractions ϕ, varying μ from 1.00 × 10− 3 Pa·s (water, ϕ = 0) to 1.41 Pa·s (pure glycerin, ϕ = 1), as listed in Table 2.
a
b ω
W=0.02m
y x
=0.2m
Fig. 2. The schematic drawing of the: (a) rotating drum; (b) tank.
S.H. Chou, S.S. Hsiau / Powder Technology 226 (2012) 99–106 Table 1 Experimental parameters: the particles are glass beads (ρb = 2476 kg/m3) with diameters of d = 4 mm. Filling degree (f):
0.2, 0.3, 0.4, 0.5
Drum speed (ω): Liquid viscosity (μ):
0.1047 rad/s, 0.2093 rad/s, 0.3140 rad/s, 0.4187 rad/s 1.00 × 10− 3 Pa·s, 2.50 × 10− 3 Pa·s, 6.00 × 10− 3 Pa·s, 3.55 × 10− 2 Pa·s, 6.01 × 10− 2 Pa·s, 1.09 × 10− 1 Pa·s, 2.19 × 10− 1 Pa·s, 5.23 × 10− 1 Pa s, 1.41 Pa·s
101
granular system behaves more like a liquid or a gas when it has a relatively higher granular temperature. The granular temperature in the ith bin in a quasi-two-dimensional system can be calculated by 2
Ti ¼
2
b u′ i þ v′i > : 2
ð5Þ
3. Results and discussion A high-speed CMOS camera (IDT X-3 plus, monochrome, capable of shooting 2000 frames per second (fps) with a resolution of 1280 × 1024 pixels) was used to record the sequential motion of the granular flows during the experiments. The flow was illuminated by two halogen lamps: DEDO DLH650 650 W 3200 K. The digital images were transported to a personal computer for further analysis. The autocorrelation technique was employed to process the stored images and decide the shift of each tracer particle in every two consecutive images [24]. In this present work, we divided the system into several sub-regions for calculation. The ensemble average velocities in each bin are averaged from about 845 tracer particles (1300 frames): Ni P
uki bui >¼ k¼1 ; Ni
ð1Þ
Ni P
vki bvi >¼ k¼1 ; Ni
ð2Þ
where bui> and bvi> denoted the ensemble average velocities in the x and y direction, respectively, in the ith bin with averaging from velocities from Ni tracer particles. The subscript k represents the kth tracer particle in the ith bin. The fluctuation velocities in the ith bin are defined as the root mean square of the deviations between the local velocities and the ensemble average velocities:
′2
1=2
bui >
′2
1=2
bvi >
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uN u Pi u ðu − bu >Þ2 ki i u t ¼ k¼1 ; Ni
ð3Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uN u Pi u ðv − bv >Þ2 ki i u t ¼ k¼1 : Ni
ð4Þ
The granular temperature T represents the specific fluctuation kinetic energy per unit mass of the granular flow because of the random motions of the particles, and can be used to quantify the kinetic fluctuation energy of the granular flow. This is a key property for studying the dynamic behavior of granular flows [15,25–27]. A Table 2 Water–glycerol mixture used in the experiments: μ is the viscosity; ρf is the density; γ is the surface tension of the fluid; and ϕ is weight percentage of glycerol to water [35]. Fluid Water Water–glycerol-1 Water–glycerol-2 Water–glycerol-3 Water–glycerol-4 Water–glycerol-5 Water–glycerol-6 Water–glycerol-7 Water–glycerol-8
ϕ 0.0 0.3 0.5 0.75 0.8 0.85 0.9 0.95 1
μ (Pa·s) −3
1.00 × 10 2.50 × 10− 3 6.00 × 10− 3 3.55 × 10− 2 6.01 × 10− 2 1.09 × 10− 1 2.19 × 10− 1 5.23 × 10− 1 1.41
ρf (kg m− 3)
γ (N m− 1)
1000 1079 1132 1198 1212 1225 1238 1251 1265
0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003
In this section, the results of the experiments are presented and discussed. The three operational parameters of this system, namely the viscosities of the liquids, filling degree of granular material and the rotation speed of rotating drum, all have an impact on the dynamic processes; the other system parameters remain fixed; see Table 1. Fig. 3(a)–(h) presents high-resolution images and the corresponding velocity field taken at different interstitial liquid viscosities, but using the same rotation speed, ω = 0.3140 rad/s. We found that four different flow regimes would occur with increasing liquid viscosity. At lower liquid viscosity ( b 6.0 mPa·s), the flow profile is characteristic of the rolling regime, Fig. 3(a)–(b). At this time, two important flow regions exist in the granular material in a rotating drum: the flowing layer region and the fixed bed region. When the rotation of the drum drives the particles into the flowing layer, they may avalanche and roll downward along the free surface. The free surface is almost flat, with an inclined angle relative to the horizontal plane, which is called the angle of repose. When particles move into the fixed bed regime, all particles would move at the same radial position. The flowing dynamic of the particles is called the plug-flow. Gray [28] proposed a general theoretical framework where the flowing layer is treated as a shallow incompressible Mohr–Coulomb or inviscid material sliding on a moving bed with erosion and deposition. The fixed bed is treated as a rigid rotating body, and the two regions are coupled together using a mass jump condition. At medium viscosity, the bed surface begins to arch and the cascading process sets in, Fig. 3(c)–(d). The height of the arch of the kidney-shaped bed increases with increasing liquid viscosity. Many large voids form due to expansion in the top-left part of the flowing layer. According to our previous study [22], if the system is a binarymixture granular system, this makes the particle percolation mechanism less selective in cascading regime. If the liquid viscosity increases continuously, the ejection of particles into the free fluid space is generally considered a required characteristic for the transition from a cascading regime to a cataracting regime, Fig. 3(e)–(f). With increasing liquid viscosity, the number of particles is thrown off, and the length of the trajectories increase until a uniform trickling veil forms along the diameter. With further increase of the liquid viscosity, particles would begin to flow with almost the same flow direction as the interstitial fluid. We call this the suspension regime, Fig. 3(g)–(h). In this regime, particles are approximately homogeneously distributed in the fluid and the interactive collisions are weaker. The angle of repose is an important parameter for understanding the physical mechanisms characterizing granular materials in a rotating drum. Several studies have been done where the segregation phenomenon is characterized by the angle of repose of the particles [29,30], and some studies also use this parameter (angle of repose) to indicate the transition of the flow regime [10,31]. There are two kinds of repose angles. One is called the static angle of repose (θs), which is the maximum angle at which the granular materials remain stationary when they are tilted. This angle depends on the frictional properties and the packing of the particles. The other is the dynamic angle of repose (θm), which is the angle between the horizontal and free surface of a granular pile after a land-slide has restored the pile to a metastable equilibrium slope. The dynamic angle is usually smaller than the static angle, because the kinetic friction between the particles is lower than
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S.H. Chou, S.S. Hsiau / Powder Technology 226 (2012) 99–106
a
a ω
103
f = 0.2 f = 0.3 f = 0.4 f = 0.5
55
50
O
θm(deg.)
α
45
40
θm
35
Rolling
b
10-3
ω
10-2
10-1
µ(Pas)
b O
55
θ1
Cascading Fig. 4. Schematic views of a cross-section of the rotating drum showing the: (a) rolling regime; (b) cascading regime.
the static friction [32]. In this study, the dynamic angle of repose is measured by superimposing 15 images of the flow in the drum after the flow has reached a steady state. Then drawing a line along the topmost particles. Fig. 4(a) shows a schematic representation of the dynamic angle of repose (θm) during the rolling flow regime. Fig. 4(b) shows the dynamic angle of repose (θm) during the cascading flow regime. In this regime, the dynamic angle of repose is not always the same along the free surface from upstream to downstream. In this study, this flow is measured in a similar way, but the position where θm arises on the line along the topmost particles is about at the middle of the rotating drum. Fig. 5(a) displays the steady-state dynamic angles of repose for different filling degrees as a function of the viscosity (ω = 0.1047 rad/s). We find that the increase of the liquid viscosity causes an increase in the angle of repose, perhaps related to the increased hydrodynamic shear force as one particle slides past another. Fig. 5(b) displays the relationship between the dynamic angle of repose and rotation speed in a slurry granular system (μ = 3.55 × 10− 2 Pa·s). The inertial force increases when the rotation speed increases, the force used to resist the gliding force resulting from gravity also increases. Therefore, we find that the angle of repose increases with increasing rotation speed. In this study, we also want to investigate the relationship between the liquid viscosity and the change of inclination angle (Δθ = θm − θ1 as shown in Fig. 4). However, if the granular flow regime is a cataracting regime or a suspension regime, these are not easily identified. Therefore, the cases where the change of inclination angle is simple to measure (rolling and cascading regime) as a function of the viscosity are shown in Fig. 6. In this figure, we find that at lower liquid viscosity, the free surface is almost flat, so that the change of inclination angle (Δθ) will be approximately 0. We call this particle flow behavior the
θm(deg.)
θm 50
45
f = 0.2 f = 0.3 f = 0.4 f = 0.5
40 0.1
0.15
0.2
0.25
ω (rad/s)
0.3
0.35
0.4
Fig. 5. Relationship between the dynamic angle of repose and (a) the liquid viscosity at ω = 0.1047 rad/s; (b) the rotation speed at μ = 3.55 × 10− 2 Pa·s. f is the filling degree.
rolling regime. When the liquid viscosity increases, the flat surface becomes deformed and S-shaped, the granular flow behavior is in the cascading regime at this time. The deformation can be approximated by two straight lines with different slopes close to this transition. According to the above discussion, the transition of the flow regime will occur when the liquid viscosity changes, so the change of inclination angle (Δθ) increases with an increase in the liquid viscosity, and the curvature also increases with increasing liquid viscosity. Furthermore, the change of inclination angle (Δθ) increases with the increasing rotation speed as well. In order to understand the distributions of particles in the drum, we track the distance dm between the center of the rotating drum and position of the centroid of all particles: N P
mi ri dm ¼ R−O ¼ i¼1N −rc ; P mi
ð6Þ
i¼1
where R is the center of mass of the system of particles; O is the center of the rotating drum; ri is the position of the ith particle; mi is the mass of the ith particle; N is the total number of particles; and rc is the position of the center of the rotating drum. This distance, ξc,
Fig. 3. High-resolution images and the corresponding velocity fields taken at different interstitial liquid viscosities, but using the same rotation speed, ω = 0.3140 rad/s: (a)–(b) μ = 1.00 × 10− 3 Pa·s, rolling regime; (c)–(d) μ = 3.55 × 10− 2 Pa·s, cascading regime; (e)–(f) μ = 2.19 × 10− 1 Pa·s, cataracting regime; (g)–(h) μ = 1.41 Pa·s, suspension regime.
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Fig. 6. Relationship between the change of inclination angle (Δθ) and the liquid viscosity. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s; ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
between the centroid of all particles and the center of the tank was non-dimensionalized by dividing it by the distance of the initial configuration, as shown in Fig. 7. When ξc is 0 this represents a homogeneous distribution of the particles; when ξc is 1 the distribution of the particles is the same as the initial configuration. The results show that the dimensionless distance ξc is close to 1 at small liquid viscosity, meaning that at lower rotation speed or liquid viscosity, the particles lie on the rotating drum as a densely packed granular bed. So the dimensionless distance is the same as the initial configuration. When the liquid viscosity is above a critical value, the dimensionless distance will decrease rapidly and approach zero at high liquid viscosity. With high rotation speed or liquid viscosity, a few particles would leave the granular bed and be injected into the pure solvent region when the liquid viscosity is above a critical value. The granular flow behavior will transform into a cataracting regime when the dimensionless distance is less than 0.89. The dimensionless distance decreases rapidly
with increasing liquid viscosity. When the liquid viscosity increased continuously, we found that the distribution of particles became more homogeneous, and the dimensionless distance was close to zero. Understanding the velocity pattern is essential to an accurate representation of the dynamics of the flow. The mean velocity is the average absolute value of the magnitude of all particle velocities taken from the entire region inside the rotating drum for all cases and is plotted as a function of the liquid viscosity in Fig. 8. We found that higher energy went into the granular system with an increase of the rotation speed. Thus the mean velocity increases with increasing rotation speed. Moreover, the results also show that the mean velocity decreases continuously with increasing liquid viscosity, but the trend becomes reversed at high liquid viscosity. The reason is that an increase in the liquid viscosity causes an increase in the fluid drag force, thus the mean velocity decreases continuously with an increase of liquid viscosity. When the liquid viscosity is above a critical value, the mean velocity will increase. In this condition, particles will begin to flow in almost the same direction as the interstitial fluid. The granular flow behavior will transform into a suspension regime. In this flow regime, the configuration of the flowing dynamic of the particles is like a plugflow, and the distribution of the particles is homogeneous. Therefore, the mean velocity will increase with increasing liquid viscosity at the same rotation speed. Furthermore, from Fig. 8, it can be seen that the mean velocity decreases linearly with increasing liquid viscosity at the same rotation speed before the trend becomes reversed at high liquid viscosity. The slopes are 3.2 × 10− 3 for a rotation speed of 0.1047 rad/s, 6.1 × 10− 3 for a rotation speed of 0.2093 rad/s, 8.8 × 10− 3 for a rotation speed of 0.3140 rad/s and 1.26 × 10− 2 for a rotation speed of 0.4187 rad/s. This can be explained as follows: the fluid viscous force plays the role of a drag force related to the avalanche of particles. At larger rotation speeds, the flowing layer thickness is thicker [20], so the number of particles in the avalanche rolling downward becomes greater. Therefore, the liquid viscosity has a greater degree of influence on the mean velocity at larger rotation speeds. The granular temperature is a key property for describing the flow of granular materials [27]. The granular temperature in a granular material flow plays the same role as the thermodynamic temperature in a gas. The magnitude of granular temperature depends on the net system energy by subtracting the dissipative energy from the energy input from external resources. The energy of the system is dissipated due to inelastic collision, fractional and cohesive effect. The average
0.055 0.05
Mean Flow Velocity (m/s)
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 10−3
10−2
10−1
100
µ (Pa s) Fig. 7. Dimensionless distance between the centroids of all particles and the center of this tank for all experimental cases. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s, ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
Fig. 8. Relationship between the mean velocity and liquid viscosity. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s, ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
S.H. Chou, S.S. Hsiau / Powder Technology 226 (2012) 99–106
Q¼
A ωD2 ; ¼ t 8
ð7Þ
where A = πD 2/8 for the half-filled fraction; t⁎ = T/2 = π/ω; and D = 2R is the diameter of p theffiffiffiffiffidrum. They indicated that for small flow rate ffi (typicallyQ ¼ Q=d gdb1), intermittent avalanches occur. Moreover, the characteristic S-shape of the free surface and essentially convex 10
x 10
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 10−3
10−2
10−1
100
µ (Pa s) Fig. 10. Phase diagram of the flow regime plotted against the flow rate per unit width and the liquid viscosity. Symbol: ○, rolling regime; ◊, cascading regime; □, cataracting regime; ☆, suspension regime.
shape of the bed-layer interface occur when Q⁎ > 1. The approximate filling area of the cross-sectional plane can be found for different filling degrees by A¼
D2 ðα− sin αÞ; 8
ð8Þ
and the residence time t⁎ for the plug flow can be defined as follows:
t ¼ T⋅
α α ¼ ; 2π ω
ð9Þ
where α is the segmental angle. Substituting Eqs. (8) and (9) into Eq. (7), the flow rate per unit width, Q, for any filling degree can be found by Q¼
−4
A ωD2 sin α : 1− ¼ t α 8
ð10Þ
The results of these experiments are collected in Fig. 10. Liquid viscosity is plotted on the abscissa axis, and the flow rate per unit width is plotted on the ordinate. Four dynamic behaviors i.e., rolling, cascading, cataracting and suspension were identified. Mellmann [34] used experimental methods to investigate the bed behavior diagrams of a monodispersed dry granular system. They also compared the experimental results with the theoretical model. They indicated that the wall friction coefficient, Froude number and filling degree are the important parameters for the transition of the granular flow regimes. However, in the current study, all particles are immersed in a fluid. We found that, except for the flow rate per unit width, the change of liquid viscosity is also a dominant parameter needed to determine the granular flow regime in the slurry-granular system.
9
Average Granular Temperature (m2/s2)
x 10−3 2.2
Q
granular temperature, averaged from the granular temperature in every bin taken from the entire region inside the rotating drum for all cases is plotted as a function of the liquid viscosity in Fig. 9. It can be seen in this figure that in the slurry granular system, the average granular temperature decreases with increasing liquid viscosity, and increases with increasing rotation speed. This is also consistent with the physical explanation. In the slurry granular system, energy dissipation due to the viscous drag force being proportional to the liquid viscosity. The particle motions and interactive collisions are mitigated due to the greater viscous force resulting from the increasing liquid viscosity. The granular temperature is indicative of the energy of the kinetic fluctuation of the particles due to the interactive collisions between them. More energy is introduced into the granular system with a higher rotation speed, resulting in stronger particle motions and interactive collisions. Moreover, from Fig. 9, we can see that the liquid viscosity also has a greater degree of influence on the average granular temperature at larger rotation speeds. According to the above discussion, at larger rotation speeds, the flowing layer is thicker. Therefore, more energy will dissipate into the surrounding area because of the fluid viscous drag force. As discussed in the preceding subsections, liquid viscosity, filling degree and rotation speed of the rotating drum all affect the dynamic behavior of the granular bed. Experiments were performed to investigate their influence on these processes, following Table 1, with liquid viscosities ranging from 1.005 × 10 − 3 to 1.41 Pa s, angular velocities ranging from 0.104 to 0.418 rad/s and filling degrees between 0.2 and 0.5. Each case typically runs for 5–10 revolutions, allowing the systems enough time to reach their steady state. A granular flow model in a dry system (with a filling degree of 0.5) has recently been proposed by GDR MiDi [33]. According to this model, the flow rate (Q) is related to the filling degree (f) and residence time (t⁎) in the plug flow. The flow rate per unit width for f = 0.5 is given by
105
8 7 6 5 4 3 2 1
4. Conclusions
0 10−3
10−2
10−1
100
µ (Pa s) Fig. 9. Relationship between the average granular temperature and liquid viscosity. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s, ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
In this study we performed experiments to examine the effects of three easily controllable parameters, namely the liquid viscosity, filling degree and rotation speed in a quasi two-dimensional slurry rotating drum. Image processing technology and particle tracking methods, along with some analysis parameters were used to distinguish between different flowing regimes.
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At lower rotation speed or liquid viscosity, the inclined plane is smooth, while the flow profile is characteristic of the rolling regime. When the liquid viscosity increases, the bed surface begins to arch and the cascading process sets in. Therefore, in this study the change of inclination angle is used to separate the rolling regime and cascading regime. When the liquid viscosity increases continuously, little particles will leave the granular bed and be injected into the pure solvent region. The dimensionless distance decreases rapidly with increasing liquid viscosity. We call this particle flow behavior the cataracting regime. When the liquid viscosity rises above a critical value, particles will begin to flow in almost the same flow direction as the interstitial fluid. The granular flow behavior is transformed into a suspension regime at this time. The granular flow behavior is strongly affected by the operational parameters. The transverse motion behavior of the granular materials in a rotating drum mentioned above can be conveniently represented in the bed behavior diagram. The diagram plots the flow rate per unit width against the liquid viscosity. It provides the users of rotating drums with the possibility to decide on the flow behavior of immersed granular bed material used, by the parameters of liquid viscosity, filling degree and rotation speed. References [1] B. Chaudhuri, F.J. Muzzio, M.S. Tomassone, Modeling of heat transfer in granular flow in rotating vessels, Chemical Engineering Science 61 (2006) 6348–6360. [2] M. Kwapinska, G. Saage, E. Tsotsas, Continuous versus discrete modeling of heat transfer to agitated beds, Powder Technology 181 (2008) 331–342. [3] J. Lehmberg, M. Hehl, K. Schugerl, Transverse mixing and heat transfer in horizontal rotary drum reactors, Powder Technology 18 (1977) 149–163. [4] H.R. Perry, C.H. Chilton, Chemical Engineers' Handbook, Vol. 6, McGraw-Hill, New York, 2003, pp. 11–46. [5] P. Lybaert, Wall-particle heat transfer in rotating heat exchangers, International Journal of Heat and Mass Transfer 30 (1987) 1663–1672. [6] D.R. Van Puyvelde, B.R. Young, M.A. Wilson, S.J. Schmidt, Experimental determination of transverse mixing kinetics in a rolling drum by image analysis, Powder Technology 106 (1999) 183–191. [7] D.V. Khakhar, A.V. Orpe, S.K. Hajra, Segregation of granular materials in rotating cylinders, Physica Acta 318 (2003) 129–136. [8] N. Jain, J.M. Ottino, R.M. Lueptow, Combined size and density segregation and mixing in noncircular tumblers, Physical Review. E 71 (2005) 051301. [9] H. Henein, J.K. Brimacombe, A.P. Watkinson, Experimental study of transverse bed motions in rotary kilns, Metal Transplantation B 14 (1983) 191–205. [10] J. Rajchenbach, Flow in powders: from discrete avalanches to continuous regime, Physical Review Letters 65 (1990) 2221–2224. [11] N. Jain, J.M. Ottino, R.M. Lueptow, An experimental study of the flowing granular layer in a rotating tumbler, Physics of Fluids 14 (2002) 572–582.
[12] A.A. Boating, P.V. Barr, Granular flow behavior in the transverse plane of a partially filled rotating cylinder, Journal of Fluid Mechanics 330 (1997) 233–249. [13] L.S. Lu, S.S. Hsiau, DEM simulation of particle mixing in a sheared granular flow, Particulogy 6 (2008) 445–454. [14] R.Y. Yang, R.P. Zou, A.B. Yu, Microdynamic analysis of particle flow in a horizontal rotating drum, Powder Technology 130 (2003) 138–146. [15] C.C. Liao, S.S. Hsiau, Influence of interstitial fluid viscosity on transport phenomenon in sheared granular materials, Chemical Engineering Science 64 (2009) 2562–2569. [16] D. Doppler, P. Gondret, T. Loiseleux, S. Meyer, M. Rabaud, Relaxation dynamics of water-immersed granular avalanches, Journal of Fluid Mechanics 577 (2007) 161–181. [17] T. Divoux, J.-C. Géminard, Friction and dilatancy in immersed granular matter, Physical Review Letters 99 (2007) 258301. [18] M. Medved, H.M. Jaeger, S.R. Nagel, Convection in a fully immersed granular slurry, Physical Review. E 63 (2001) 061302. [19] S.J. Fiedor, J.M. Ottion, Dynamics of axial segregation and coarsening of dry granular materials and slurries in circular and square tubes, Physical Review Letters 91 (2003) 244301. [20] N. Jain, J.M. Ottino, R.M. Lueptow, Effect of interstitial fluid on a granular flowing layer, Journal of Fluid Mechanics 508 (2004) 23–44/sb:pages>. [21] T. Finger, R. Stannarius, Influences of the interstitial liquid on segregation patterns of granular slurries in a rotating drum, Physical Review. E 75 (2007) 301308. [22] C.C. Liao, S.S. Hsiau, K.W. To, Granular dynamics of a slurry in a rotating drum, Physical Review. E 82 (2010) 010302. [23] S.H. Chou, C.C. Liao, S.S. Hsiau, The effect of interstitial fluid viscosity on particle segregation in a slurry rotating drum, Physics of Fluids 23 (2011) 083301. [24] S.S. Hsiau, H.W. Jang, Measurements of velocity fluctuations of granular materials in a shear cell, Experimental Thermal and Fluid Science 17 (1998) 202–209. [25] G. Bunin, Y. Shokef, D. Levine, Frequency-dependent fluctuation–dissipation relations in granular gases, Physical Review. E 77 (2008) 051301. [26] C.S. Campbell, Rapid granular flows, Annual Review of Fluid Mechanics 22 (1990) 57–92. [27] S. Ogawa, Multi-temperature theory of granular materials, Proceedings of US–Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials (Tokyo), 208, 1978. [28] J.M.N.T. Gray, Granular flow in partially filled slowly rotating drums, Journal of Fluid Mechanics 441 (2001) 1–29. [29] S.H. Chou, C.C. Liao, S.S. Hsiau, An experimental study on the effect of liquid content and viscosity on particle segregation in a rotating drum, Powder Technology 201 (2010) 266–272. [30] K.M. Hill, J. Kakalios, Reversible axial segregation of rotating granular media, Physical Review. E 52 (1995) 4393–4400. [31] R. Fischer, P. Gondret, M. Rabaud, Transition by intermittency in granular matter: from discontinuous avalanches to continuous flow, Physical Review Letters 103 (2009) 128002. [32] H.M. Jaeger, C.H. Liu, S.R. Nagel, T.A. Witten, Friction in granular flows, European Letters 11 (1990) 619–624. [33] G.D.R. MiDi, On dense granular flow, The European Physical Journal E 14 (2004) 341–365. [34] J. Mellmann, The transverse motion of solids in rotating cylinders-forms of motion and transition behavior, Powder Technology 118 (2001) 251–270. [35] N.E. Dorsey, Properties of Ordinary Water-substance in All Its Phases: Water Vapor, Water, and All the Ices, 7nd ed. Reinhold Pub. Corp., New York, 1940.
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Dynamic properties of immersed granular matter in different flow regimes in a rotating drum S.H. Chou, S.S. Hsiau ⁎ Department of Mechanical Engineering, National Central University, Jhongli 32001, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 30 January 2012 Received in revised form 26 March 2012 Accepted 12 April 2012 Available online 21 April 2012 Keywords: Rotating drum Rotation speed Filling degree Liquid viscosity Flow regime
a b s t r a c t Using particle tracking velocimetry, we study the dynamic properties and flowing behavior of immersed granular matter in a rotating drum. In this study, the interstitial fluid is water or a water–glycerol mixture. The filling degree of the particles, the rotation speed, and the viscosity of the interstitial fluid are the three experimental control parameters. The results show that both the granular dynamic properties and flowing behavior are strongly affected by the operational parameters. At lower rotation speed or liquid viscosity, the distance between the centroid of all particles and the center of the tank is the same as the initial configuration, but the distance decreases rapidly when the liquid viscosity is above a critical value. The mean velocity, obtained by averaging the velocities of all particles will decrease with the increase of the liquid viscosity. When the liquid viscosity is above a critical value, the mean velocity will increase, and the granular flow behavior will transform into a suspension regime. Furthermore, the experimental results indicate that the liquid viscosity and the flow rate per unit width have a significant influence on the dynamic properties and flow behavior of the immersed granular matter. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Rotating drums are usually simply cylinders rotating about its central axis so as to drive particle motion. Devices based on this configuration are widely used for processing granular materials in the mineral, pharmaceutical, ceramic, cement and food industries, in which they are used to perform drying, heating [1,2], chemical reactions [3–5], mixing and segregation [6–8]. Although the devices are simple and can be operated relatively easily, granular dynamic behavior is more complicated. Granular flow behavior in a rotating drum can be identified by several flow regimes based on the particle motion. The particle flow behavior and mixing/segregation mechanisms may be different in each flow regime. Six identifiable flow regimes in a dry granular system may be used to describe the particle motion in a rotating drum depending on different operational conditions, including the rotational speed, wall friction coefficient, filling degree, and so on. The flow regimes include slipping, slumping, rolling, cascading, cataracting and centrifuging [9,10], as shown in Fig. 1. However in the past, most studies have been on dry granular systems, where the interstitial fluid is air. It has been found that the investigation of particle transport properties is about the influence of different particle materials, size of particles or tank, even the rotation speed of the rotating drum, while the fluid (air) effects are small enough to be neglected. Jain et al. [11] used particle tracking
⁎ 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 © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2012.04.024
velocimetry to investigate the velocity profile in the fluidized layer with different particle sizes. They found that the streamwise velocity profile is nearly linear near the top of the flowing layer, while close to the interface between the flowing layer and fixed bed the velocity logarithmically decreases to zero. Furthermore, near the surface of the flowing layer, the velocity fluctuations can be as large as one-third of the maximum streamwise velocity. Boating and Barr [12] used optical-fiber probes to measure the mean velocities with different particle materials in different sizes of rotating drum. The results show that the bulk viscosity is proportional to the shear viscosity and dilation and inversely proportional to particle size. Most computer simulation studies have also focused on dry granular systems. Lu and Hsiau [13] used the discrete element method (DEM) to investigate the mixing behavior of particles in a sheared granular flow. They found that the mixing layer thickness is greater, and the mixing growth rate is faster in the upper part of the test section. Yang et al. [14] used DEM to investigate the microdynamic variables, including the porosity, coordination number, collision velocity and collision frequency in the rotating drum. In the past few years, some researchers have begun to focus on investigating the effects of liquid viscosity on particle flowing behavior in a slurry granular system [15–17]. When the interstitial fluid is liquid, the fluid (liquid) effect cannot be neglected. Medved et al. [18] investigated the convection in a granular system completely immersed in fluid inside a vertical cylindrical container. In comparison with a dry system, they found that the wet system flows up to two orders of magnitude more slowly than the dry one dose. Fiedor and Ottion [19] studied the dynamics of axial segregation and coarsening of dry
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Slipping
Slumping
Cascading
Rolling
Cataracting
Centrifuging
Fig. 1. Forms of transverse motion of solids in a rotating drum.
granular materials and slurries in tubes. They indicated that the fraction of surface area of small-rich particles increases resulting from the increasing viscosity of interstitial fluid. Jain et al. [20] used particle tracking velocimetry to measure the velocity field in a slurry system with different Froude numbers, bead sizes, fluid densities and fluid viscosity. They found that the thickness of the flowing layer and angle of repose with liquid as the interstitial fluid are generally larger than in the dry system with similar conditions, but the shear rate is generally smaller. Finger and Stannarius [21] investigated the influence of the viscosity of the interstitial liquid in a horizontally rotating mixer. They found that the viscosity of interstitial liquid played a crucial role in pattern dynamics and the structure of the segregation patterns. They also indicated that the density effect of interstitial fluid dose not influence the formation and evolution of segregation patterns of the granular materials. Liao et al. [22] studied the effects of interstitial fluid viscosity on the rates of dynamic processes in a thin rotating drum. They indicated that the characteristic speed of a bead in the flowing layer decreases with the fluid viscosity, but the mixing rate of beads is found to increase with fluid viscosity. Chou et al. [23] investigated the phenomena of particle segregation and flowing behavior in a slurry rotating drum with liquids having different viscosities and different filling degrees. They indicated that the segregation index and angle of repose are shown to decrease with increased liquid viscosity. When the liquid viscosity is the same, the increase in the filling degree causes the segregation index to increase, while the net rate of mixing seems to decrease. We found that the fluid kinematic viscosity is a very important parameter determining the flow behavior if the particles are fully immersed in liquid, but many unknown physical mechanisms exist. Therefore, in this study, we use experimental methods to obtain the dynamic properties of immersed granular matter in a quasi-twodimensional slurry rotating drum, where the interstitial fluid is water or a glycerol–water mixture. Our intent is to determine the macrodynamic variables, including the particle mean velocity, average granular temperature, and flowing behavior through particle tracking velocimetry. The filling degree of particles, the rotational speed and the density and viscosity of the interstitial fluid are different. 2. Experimental procedure A schematic representation of the circular drum used in the quasitwo-dimensional experiments is shown in Fig. 2. The diameter of the drum is 0.2 m and the gap width W is 0.02 m. The rear surface of the drum was constructed of a black anodized aluminum plate to minimize electrostatic effects on the particles and optical noise effects in the digital images. The front faceplate was made of clear acrylic to permit optical access. A small hole in the side of the tank permitted liquid to be injected from a hopper. A stepper motor and micro series driver combination was used to rotate the drum at a speed of 0.1047 rad/s, 0.2093 rad/s, 0.3140 rad/s, and 0.4187 rad/s, corresponding to the Froude number, Fr = Rω 2/g of 1.67 × 10 − 4, 6.71 × 10 − 4, 1.51 × 10 − 3, and 2.68 × 10 − 3, where ω is the angular velocity of the drum (=2π/T, T = rotation period); R is the radius of the drum; and g is the acceleration of gravity. The dimensionless axial
thickness of the drum, defined as the ratio of the drum's axial length and the big particle diameter, was set to 5 in this study. In each experiment, mono-sized glass beads were used as the granular material. The diameter of the glass beads was 4 mm with a standard deviation of 0.09 mm and their density ρ was 2.5 g/cm 3. Details of the experimental conditions are provided in Table 1. In all of the slurry flow experiments, the drum was completely filled (no air content) with a water–glycerin mixture before it is sealed. Some experiments were carried out with pure water as the interstitial fluid. In addition, several experiments were performed with mixtures of water and glycerin in a range of different viscosities. We use water–glycerin mixtures with different glycerin weight fractions ϕ, varying μ from 1.00 × 10− 3 Pa·s (water, ϕ = 0) to 1.41 Pa·s (pure glycerin, ϕ = 1), as listed in Table 2.
a
b ω
W=0.02m
y x
=0.2m
Fig. 2. The schematic drawing of the: (a) rotating drum; (b) tank.
S.H. Chou, S.S. Hsiau / Powder Technology 226 (2012) 99–106 Table 1 Experimental parameters: the particles are glass beads (ρb = 2476 kg/m3) with diameters of d = 4 mm. Filling degree (f):
0.2, 0.3, 0.4, 0.5
Drum speed (ω): Liquid viscosity (μ):
0.1047 rad/s, 0.2093 rad/s, 0.3140 rad/s, 0.4187 rad/s 1.00 × 10− 3 Pa·s, 2.50 × 10− 3 Pa·s, 6.00 × 10− 3 Pa·s, 3.55 × 10− 2 Pa·s, 6.01 × 10− 2 Pa·s, 1.09 × 10− 1 Pa·s, 2.19 × 10− 1 Pa·s, 5.23 × 10− 1 Pa s, 1.41 Pa·s
101
granular system behaves more like a liquid or a gas when it has a relatively higher granular temperature. The granular temperature in the ith bin in a quasi-two-dimensional system can be calculated by 2
Ti ¼
2
b u′ i þ v′i > : 2
ð5Þ
3. Results and discussion A high-speed CMOS camera (IDT X-3 plus, monochrome, capable of shooting 2000 frames per second (fps) with a resolution of 1280 × 1024 pixels) was used to record the sequential motion of the granular flows during the experiments. The flow was illuminated by two halogen lamps: DEDO DLH650 650 W 3200 K. The digital images were transported to a personal computer for further analysis. The autocorrelation technique was employed to process the stored images and decide the shift of each tracer particle in every two consecutive images [24]. In this present work, we divided the system into several sub-regions for calculation. The ensemble average velocities in each bin are averaged from about 845 tracer particles (1300 frames): Ni P
uki bui >¼ k¼1 ; Ni
ð1Þ
Ni P
vki bvi >¼ k¼1 ; Ni
ð2Þ
where bui> and bvi> denoted the ensemble average velocities in the x and y direction, respectively, in the ith bin with averaging from velocities from Ni tracer particles. The subscript k represents the kth tracer particle in the ith bin. The fluctuation velocities in the ith bin are defined as the root mean square of the deviations between the local velocities and the ensemble average velocities:
′2
1=2
bui >
′2
1=2
bvi >
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uN u Pi u ðu − bu >Þ2 ki i u t ¼ k¼1 ; Ni
ð3Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uN u Pi u ðv − bv >Þ2 ki i u t ¼ k¼1 : Ni
ð4Þ
The granular temperature T represents the specific fluctuation kinetic energy per unit mass of the granular flow because of the random motions of the particles, and can be used to quantify the kinetic fluctuation energy of the granular flow. This is a key property for studying the dynamic behavior of granular flows [15,25–27]. A Table 2 Water–glycerol mixture used in the experiments: μ is the viscosity; ρf is the density; γ is the surface tension of the fluid; and ϕ is weight percentage of glycerol to water [35]. Fluid Water Water–glycerol-1 Water–glycerol-2 Water–glycerol-3 Water–glycerol-4 Water–glycerol-5 Water–glycerol-6 Water–glycerol-7 Water–glycerol-8
ϕ 0.0 0.3 0.5 0.75 0.8 0.85 0.9 0.95 1
μ (Pa·s) −3
1.00 × 10 2.50 × 10− 3 6.00 × 10− 3 3.55 × 10− 2 6.01 × 10− 2 1.09 × 10− 1 2.19 × 10− 1 5.23 × 10− 1 1.41
ρf (kg m− 3)
γ (N m− 1)
1000 1079 1132 1198 1212 1225 1238 1251 1265
0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003 0.07 ± 0.003
In this section, the results of the experiments are presented and discussed. The three operational parameters of this system, namely the viscosities of the liquids, filling degree of granular material and the rotation speed of rotating drum, all have an impact on the dynamic processes; the other system parameters remain fixed; see Table 1. Fig. 3(a)–(h) presents high-resolution images and the corresponding velocity field taken at different interstitial liquid viscosities, but using the same rotation speed, ω = 0.3140 rad/s. We found that four different flow regimes would occur with increasing liquid viscosity. At lower liquid viscosity ( b 6.0 mPa·s), the flow profile is characteristic of the rolling regime, Fig. 3(a)–(b). At this time, two important flow regions exist in the granular material in a rotating drum: the flowing layer region and the fixed bed region. When the rotation of the drum drives the particles into the flowing layer, they may avalanche and roll downward along the free surface. The free surface is almost flat, with an inclined angle relative to the horizontal plane, which is called the angle of repose. When particles move into the fixed bed regime, all particles would move at the same radial position. The flowing dynamic of the particles is called the plug-flow. Gray [28] proposed a general theoretical framework where the flowing layer is treated as a shallow incompressible Mohr–Coulomb or inviscid material sliding on a moving bed with erosion and deposition. The fixed bed is treated as a rigid rotating body, and the two regions are coupled together using a mass jump condition. At medium viscosity, the bed surface begins to arch and the cascading process sets in, Fig. 3(c)–(d). The height of the arch of the kidney-shaped bed increases with increasing liquid viscosity. Many large voids form due to expansion in the top-left part of the flowing layer. According to our previous study [22], if the system is a binarymixture granular system, this makes the particle percolation mechanism less selective in cascading regime. If the liquid viscosity increases continuously, the ejection of particles into the free fluid space is generally considered a required characteristic for the transition from a cascading regime to a cataracting regime, Fig. 3(e)–(f). With increasing liquid viscosity, the number of particles is thrown off, and the length of the trajectories increase until a uniform trickling veil forms along the diameter. With further increase of the liquid viscosity, particles would begin to flow with almost the same flow direction as the interstitial fluid. We call this the suspension regime, Fig. 3(g)–(h). In this regime, particles are approximately homogeneously distributed in the fluid and the interactive collisions are weaker. The angle of repose is an important parameter for understanding the physical mechanisms characterizing granular materials in a rotating drum. Several studies have been done where the segregation phenomenon is characterized by the angle of repose of the particles [29,30], and some studies also use this parameter (angle of repose) to indicate the transition of the flow regime [10,31]. There are two kinds of repose angles. One is called the static angle of repose (θs), which is the maximum angle at which the granular materials remain stationary when they are tilted. This angle depends on the frictional properties and the packing of the particles. The other is the dynamic angle of repose (θm), which is the angle between the horizontal and free surface of a granular pile after a land-slide has restored the pile to a metastable equilibrium slope. The dynamic angle is usually smaller than the static angle, because the kinetic friction between the particles is lower than
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S.H. Chou, S.S. Hsiau / Powder Technology 226 (2012) 99–106
a
a ω
103
f = 0.2 f = 0.3 f = 0.4 f = 0.5
55
50
O
θm(deg.)
α
45
40
θm
35
Rolling
b
10-3
ω
10-2
10-1
µ(Pas)
b O
55
θ1
Cascading Fig. 4. Schematic views of a cross-section of the rotating drum showing the: (a) rolling regime; (b) cascading regime.
the static friction [32]. In this study, the dynamic angle of repose is measured by superimposing 15 images of the flow in the drum after the flow has reached a steady state. Then drawing a line along the topmost particles. Fig. 4(a) shows a schematic representation of the dynamic angle of repose (θm) during the rolling flow regime. Fig. 4(b) shows the dynamic angle of repose (θm) during the cascading flow regime. In this regime, the dynamic angle of repose is not always the same along the free surface from upstream to downstream. In this study, this flow is measured in a similar way, but the position where θm arises on the line along the topmost particles is about at the middle of the rotating drum. Fig. 5(a) displays the steady-state dynamic angles of repose for different filling degrees as a function of the viscosity (ω = 0.1047 rad/s). We find that the increase of the liquid viscosity causes an increase in the angle of repose, perhaps related to the increased hydrodynamic shear force as one particle slides past another. Fig. 5(b) displays the relationship between the dynamic angle of repose and rotation speed in a slurry granular system (μ = 3.55 × 10− 2 Pa·s). The inertial force increases when the rotation speed increases, the force used to resist the gliding force resulting from gravity also increases. Therefore, we find that the angle of repose increases with increasing rotation speed. In this study, we also want to investigate the relationship between the liquid viscosity and the change of inclination angle (Δθ = θm − θ1 as shown in Fig. 4). However, if the granular flow regime is a cataracting regime or a suspension regime, these are not easily identified. Therefore, the cases where the change of inclination angle is simple to measure (rolling and cascading regime) as a function of the viscosity are shown in Fig. 6. In this figure, we find that at lower liquid viscosity, the free surface is almost flat, so that the change of inclination angle (Δθ) will be approximately 0. We call this particle flow behavior the
θm(deg.)
θm 50
45
f = 0.2 f = 0.3 f = 0.4 f = 0.5
40 0.1
0.15
0.2
0.25
ω (rad/s)
0.3
0.35
0.4
Fig. 5. Relationship between the dynamic angle of repose and (a) the liquid viscosity at ω = 0.1047 rad/s; (b) the rotation speed at μ = 3.55 × 10− 2 Pa·s. f is the filling degree.
rolling regime. When the liquid viscosity increases, the flat surface becomes deformed and S-shaped, the granular flow behavior is in the cascading regime at this time. The deformation can be approximated by two straight lines with different slopes close to this transition. According to the above discussion, the transition of the flow regime will occur when the liquid viscosity changes, so the change of inclination angle (Δθ) increases with an increase in the liquid viscosity, and the curvature also increases with increasing liquid viscosity. Furthermore, the change of inclination angle (Δθ) increases with the increasing rotation speed as well. In order to understand the distributions of particles in the drum, we track the distance dm between the center of the rotating drum and position of the centroid of all particles: N P
mi ri dm ¼ R−O ¼ i¼1N −rc ; P mi
ð6Þ
i¼1
where R is the center of mass of the system of particles; O is the center of the rotating drum; ri is the position of the ith particle; mi is the mass of the ith particle; N is the total number of particles; and rc is the position of the center of the rotating drum. This distance, ξc,
Fig. 3. High-resolution images and the corresponding velocity fields taken at different interstitial liquid viscosities, but using the same rotation speed, ω = 0.3140 rad/s: (a)–(b) μ = 1.00 × 10− 3 Pa·s, rolling regime; (c)–(d) μ = 3.55 × 10− 2 Pa·s, cascading regime; (e)–(f) μ = 2.19 × 10− 1 Pa·s, cataracting regime; (g)–(h) μ = 1.41 Pa·s, suspension regime.
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Fig. 6. Relationship between the change of inclination angle (Δθ) and the liquid viscosity. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s; ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
between the centroid of all particles and the center of the tank was non-dimensionalized by dividing it by the distance of the initial configuration, as shown in Fig. 7. When ξc is 0 this represents a homogeneous distribution of the particles; when ξc is 1 the distribution of the particles is the same as the initial configuration. The results show that the dimensionless distance ξc is close to 1 at small liquid viscosity, meaning that at lower rotation speed or liquid viscosity, the particles lie on the rotating drum as a densely packed granular bed. So the dimensionless distance is the same as the initial configuration. When the liquid viscosity is above a critical value, the dimensionless distance will decrease rapidly and approach zero at high liquid viscosity. With high rotation speed or liquid viscosity, a few particles would leave the granular bed and be injected into the pure solvent region when the liquid viscosity is above a critical value. The granular flow behavior will transform into a cataracting regime when the dimensionless distance is less than 0.89. The dimensionless distance decreases rapidly
with increasing liquid viscosity. When the liquid viscosity increased continuously, we found that the distribution of particles became more homogeneous, and the dimensionless distance was close to zero. Understanding the velocity pattern is essential to an accurate representation of the dynamics of the flow. The mean velocity is the average absolute value of the magnitude of all particle velocities taken from the entire region inside the rotating drum for all cases and is plotted as a function of the liquid viscosity in Fig. 8. We found that higher energy went into the granular system with an increase of the rotation speed. Thus the mean velocity increases with increasing rotation speed. Moreover, the results also show that the mean velocity decreases continuously with increasing liquid viscosity, but the trend becomes reversed at high liquid viscosity. The reason is that an increase in the liquid viscosity causes an increase in the fluid drag force, thus the mean velocity decreases continuously with an increase of liquid viscosity. When the liquid viscosity is above a critical value, the mean velocity will increase. In this condition, particles will begin to flow in almost the same direction as the interstitial fluid. The granular flow behavior will transform into a suspension regime. In this flow regime, the configuration of the flowing dynamic of the particles is like a plugflow, and the distribution of the particles is homogeneous. Therefore, the mean velocity will increase with increasing liquid viscosity at the same rotation speed. Furthermore, from Fig. 8, it can be seen that the mean velocity decreases linearly with increasing liquid viscosity at the same rotation speed before the trend becomes reversed at high liquid viscosity. The slopes are 3.2 × 10− 3 for a rotation speed of 0.1047 rad/s, 6.1 × 10− 3 for a rotation speed of 0.2093 rad/s, 8.8 × 10− 3 for a rotation speed of 0.3140 rad/s and 1.26 × 10− 2 for a rotation speed of 0.4187 rad/s. This can be explained as follows: the fluid viscous force plays the role of a drag force related to the avalanche of particles. At larger rotation speeds, the flowing layer thickness is thicker [20], so the number of particles in the avalanche rolling downward becomes greater. Therefore, the liquid viscosity has a greater degree of influence on the mean velocity at larger rotation speeds. The granular temperature is a key property for describing the flow of granular materials [27]. The granular temperature in a granular material flow plays the same role as the thermodynamic temperature in a gas. The magnitude of granular temperature depends on the net system energy by subtracting the dissipative energy from the energy input from external resources. The energy of the system is dissipated due to inelastic collision, fractional and cohesive effect. The average
0.055 0.05
Mean Flow Velocity (m/s)
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 10−3
10−2
10−1
100
µ (Pa s) Fig. 7. Dimensionless distance between the centroids of all particles and the center of this tank for all experimental cases. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s, ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
Fig. 8. Relationship between the mean velocity and liquid viscosity. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s, ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
S.H. Chou, S.S. Hsiau / Powder Technology 226 (2012) 99–106
Q¼
A ωD2 ; ¼ t 8
ð7Þ
where A = πD 2/8 for the half-filled fraction; t⁎ = T/2 = π/ω; and D = 2R is the diameter of p theffiffiffiffiffidrum. They indicated that for small flow rate ffi (typicallyQ ¼ Q=d gdb1), intermittent avalanches occur. Moreover, the characteristic S-shape of the free surface and essentially convex 10
x 10
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 10−3
10−2
10−1
100
µ (Pa s) Fig. 10. Phase diagram of the flow regime plotted against the flow rate per unit width and the liquid viscosity. Symbol: ○, rolling regime; ◊, cascading regime; □, cataracting regime; ☆, suspension regime.
shape of the bed-layer interface occur when Q⁎ > 1. The approximate filling area of the cross-sectional plane can be found for different filling degrees by A¼
D2 ðα− sin αÞ; 8
ð8Þ
and the residence time t⁎ for the plug flow can be defined as follows:
t ¼ T⋅
α α ¼ ; 2π ω
ð9Þ
where α is the segmental angle. Substituting Eqs. (8) and (9) into Eq. (7), the flow rate per unit width, Q, for any filling degree can be found by Q¼
−4
A ωD2 sin α : 1− ¼ t α 8
ð10Þ
The results of these experiments are collected in Fig. 10. Liquid viscosity is plotted on the abscissa axis, and the flow rate per unit width is plotted on the ordinate. Four dynamic behaviors i.e., rolling, cascading, cataracting and suspension were identified. Mellmann [34] used experimental methods to investigate the bed behavior diagrams of a monodispersed dry granular system. They also compared the experimental results with the theoretical model. They indicated that the wall friction coefficient, Froude number and filling degree are the important parameters for the transition of the granular flow regimes. However, in the current study, all particles are immersed in a fluid. We found that, except for the flow rate per unit width, the change of liquid viscosity is also a dominant parameter needed to determine the granular flow regime in the slurry-granular system.
9
Average Granular Temperature (m2/s2)
x 10−3 2.2
Q
granular temperature, averaged from the granular temperature in every bin taken from the entire region inside the rotating drum for all cases is plotted as a function of the liquid viscosity in Fig. 9. It can be seen in this figure that in the slurry granular system, the average granular temperature decreases with increasing liquid viscosity, and increases with increasing rotation speed. This is also consistent with the physical explanation. In the slurry granular system, energy dissipation due to the viscous drag force being proportional to the liquid viscosity. The particle motions and interactive collisions are mitigated due to the greater viscous force resulting from the increasing liquid viscosity. The granular temperature is indicative of the energy of the kinetic fluctuation of the particles due to the interactive collisions between them. More energy is introduced into the granular system with a higher rotation speed, resulting in stronger particle motions and interactive collisions. Moreover, from Fig. 9, we can see that the liquid viscosity also has a greater degree of influence on the average granular temperature at larger rotation speeds. According to the above discussion, at larger rotation speeds, the flowing layer is thicker. Therefore, more energy will dissipate into the surrounding area because of the fluid viscous drag force. As discussed in the preceding subsections, liquid viscosity, filling degree and rotation speed of the rotating drum all affect the dynamic behavior of the granular bed. Experiments were performed to investigate their influence on these processes, following Table 1, with liquid viscosities ranging from 1.005 × 10 − 3 to 1.41 Pa s, angular velocities ranging from 0.104 to 0.418 rad/s and filling degrees between 0.2 and 0.5. Each case typically runs for 5–10 revolutions, allowing the systems enough time to reach their steady state. A granular flow model in a dry system (with a filling degree of 0.5) has recently been proposed by GDR MiDi [33]. According to this model, the flow rate (Q) is related to the filling degree (f) and residence time (t⁎) in the plug flow. The flow rate per unit width for f = 0.5 is given by
105
8 7 6 5 4 3 2 1
4. Conclusions
0 10−3
10−2
10−1
100
µ (Pa s) Fig. 9. Relationship between the average granular temperature and liquid viscosity. Symbol: ●, ω = 0.1047 rad/s; ♦, ω = 0.2093 rad/s, ■, ω = 0.3140 rad/s; ★, ω = 0.4187 rad/s. Symbol fill: open, f = 0.2; open with a ‘+’ in the middle, f = 0.3; black, f = 0.4; open with a ‘×’ in the middle, f = 0.5.
In this study we performed experiments to examine the effects of three easily controllable parameters, namely the liquid viscosity, filling degree and rotation speed in a quasi two-dimensional slurry rotating drum. Image processing technology and particle tracking methods, along with some analysis parameters were used to distinguish between different flowing regimes.
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At lower rotation speed or liquid viscosity, the inclined plane is smooth, while the flow profile is characteristic of the rolling regime. When the liquid viscosity increases, the bed surface begins to arch and the cascading process sets in. Therefore, in this study the change of inclination angle is used to separate the rolling regime and cascading regime. When the liquid viscosity increases continuously, little particles will leave the granular bed and be injected into the pure solvent region. The dimensionless distance decreases rapidly with increasing liquid viscosity. We call this particle flow behavior the cataracting regime. When the liquid viscosity rises above a critical value, particles will begin to flow in almost the same flow direction as the interstitial fluid. The granular flow behavior is transformed into a suspension regime at this time. The granular flow behavior is strongly affected by the operational parameters. The transverse motion behavior of the granular materials in a rotating drum mentioned above can be conveniently represented in the bed behavior diagram. The diagram plots the flow rate per unit width against the liquid viscosity. It provides the users of rotating drums with the possibility to decide on the flow behavior of immersed granular bed material used, by the parameters of liquid viscosity, filling degree and rotation speed. References [1] B. Chaudhuri, F.J. Muzzio, M.S. Tomassone, Modeling of heat transfer in granular flow in rotating vessels, Chemical Engineering Science 61 (2006) 6348–6360. [2] M. Kwapinska, G. Saage, E. Tsotsas, Continuous versus discrete modeling of heat transfer to agitated beds, Powder Technology 181 (2008) 331–342. [3] J. Lehmberg, M. Hehl, K. Schugerl, Transverse mixing and heat transfer in horizontal rotary drum reactors, Powder Technology 18 (1977) 149–163. [4] H.R. Perry, C.H. Chilton, Chemical Engineers' Handbook, Vol. 6, McGraw-Hill, New York, 2003, pp. 11–46. [5] P. Lybaert, Wall-particle heat transfer in rotating heat exchangers, International Journal of Heat and Mass Transfer 30 (1987) 1663–1672. [6] D.R. Van Puyvelde, B.R. Young, M.A. Wilson, S.J. Schmidt, Experimental determination of transverse mixing kinetics in a rolling drum by image analysis, Powder Technology 106 (1999) 183–191. [7] D.V. Khakhar, A.V. Orpe, S.K. Hajra, Segregation of granular materials in rotating cylinders, Physica Acta 318 (2003) 129–136. [8] N. Jain, J.M. Ottino, R.M. Lueptow, Combined size and density segregation and mixing in noncircular tumblers, Physical Review. E 71 (2005) 051301. [9] H. Henein, J.K. Brimacombe, A.P. Watkinson, Experimental study of transverse bed motions in rotary kilns, Metal Transplantation B 14 (1983) 191–205. [10] J. Rajchenbach, Flow in powders: from discrete avalanches to continuous regime, Physical Review Letters 65 (1990) 2221–2224. [11] N. Jain, J.M. Ottino, R.M. Lueptow, An experimental study of the flowing granular layer in a rotating tumbler, Physics of Fluids 14 (2002) 572–582.
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