Numerical investigation of granular flow similarity in rotating drums

Particuology 22 (2015) 119–127

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Numerical investigation of granular flow similarity in rotating drums Huabiao Qi a,b , Ji Xu a , Guangzheng Zhou a,∗ , Feiguo Chen a , Wei Ge a,∗ , Jinghai Li a a b

State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e

i n f o

Article history: Received 11 March 2014 Received in revised form 29 September 2014 Accepted 20 October 2014 Keywords: Discrete element method Granular flow Similarity Rotating drum Scale up Graphics processing unit

a b s t r a c t The theory of flow similarity has not been well established for granular flows, in contrast to the case for conventional fluids, owing to a lack of reliable and general constitutive laws for their continuum description. A rigorous investigation of the similarity of velocity fields in different granular systems would be valuable to theoretical studies. However, experimental measurements face technological and physical problems. Numerical simulations that employ the discrete element method (DEM) may be an alternative to experiments by providing similar results, where quantitative analysis could be implemented with virtually no limitation. In this study, the similarity of velocity fields is investigated for the rolling regime of rotating drums by conducting simulations based on the DEM and using graphics processing units. For a constant Froude number, it is found that the particle-to-drum size ratio plays a dominant role in the determination of the velocity field, while the velocity field is much more sensitive to some material properties than to others. The implications of these findings are discussed in terms of establishing theoretical similarity laws for granular flows. © 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Introduction Rotating drums are employed in a rich variety of industrial practices to enhance the mixing, aggregation, and heat transfer of granular materials (Zhu, Zhou, Yang, & Yu, 2008; Liu, Ge, Xiao, & Li, 2008; Xu, Xu, Zhou, Du, & Hu, 2010). Six flow regimes are commonly recognized under different rotational speeds, while the rolling regime (characterized by an upper flowing layer and lower plug flow) is frequently encountered in industrial applications (Chou & Lee, 2009). Essentially, the rolling regime falls into the category of so-called “dense granular flows”, where there are enduring force chains between adjacent particles (MiDi, 2004; Campbell, 2006; Sun, Jin, Liu, & Zhang, 2010). The dense granular flows generally exhibit rather complicated rheological behavior, and the scaling laws of rotating drums are thus still not well established owing to the lack of general and reliable constitutive laws (Forterre & Pouliquen, 2008; Ji & Shen, 2008; Zhang, Behringer, & Goldhirsch, 2010). These disadvantages usually result in large discrepancy between theoretical predictions and actual behavior in the scale-up of industrial equipment. Consequently, an in-depth investigation of the scaling behavior of rotating drums is important

∗ Corresponding authors. Tel.: +86 10 82544943; fax: +86 10 62558065. E-mail addresses: [email protected] (G. Zhou), [email protected] (W. Ge).

for understanding the physical mechanisms of granular flows and the optimization of operating conditions. To investigate the general scaling behavior of all flow regimes in horizontal rotating drums, a series of dimensionless numbers have been identified (Ding, Forster, Seville, & Parker, 2001; Mellmann, 2001), including the filling level, Froude number (Fr), ratio of particle diameter to drum diameter (referred to as the “size ratio” hereafter), and material properties. By combining fundamental dimensionless numbers, several new dimensionless groups have also been proposed for specific conditions (Alexander, Shinbrot, & Muzzio, 2002; Chou & Lee, 2009). In regard to the rolling regime, an experimental study employing flow visualization demonstrated the critical roles of the Froude number and size ratio in determining the profile and thickness of the upper flowing layer, while the dynamic angle of repose was found to depend on the material properties (Orpe & Khakhar, 2001). Liu, Specht, Gonzalez, and Walzel (2006) further revealed that the material properties affect the maximum thickness and mean particle velocity of the upper flowing layer. Additionally, it was reported that the surface velocity has different relations with the Froude number depending on the rotational speed (Alexander et al., 2002), and the relationship between the mean velocity and thickness of the flowing layer has different forms depending on the size ratio (Félix, Falk, & D’Ortona, 2007). In summary, the scaling behavior in the rolling regime of rotating drums is rather complicated, and the flow details (such as

http://dx.doi.org/10.1016/j.partic.2014.10.012 1674-2001/© 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

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the velocity profile) have not yet been systematically investigated in experiments owing to the limitations of current technologies. As a numerical approach, the discrete element method (DEM) (Cundall & Strack, 1979; Zhu, Zhou, Yang, & Yu, 2007) has notable advantages in the description of velocities of granular materials, where the trajectories of all particles are explicitly tracked. Moreover, the material properties can be adjusted much more easily than in experiments. The DEM is thus a powerful tool for the investigation of scaling behavior in granular flows. Apart from the classical linear contact model in which interparticle forces are simplified, there already exist various nonlinear models, which are theoretically more accurate but also computationally more intensive. Fortunately, general-purpose graphics processing units (GPUs) can effectively alleviate this burden to some extent, and have been increasingly applied to various numerical problems (Xu et al., 2011a; Ren et al., 2013). In this work, with a nonlinear contact model implemented on a GPU platform, the velocity fields in three-dimensional (3D) horizontal rotating drums were explored extensively for various size ratios and material properties. At a fixed filling level and Froude number in the rolling regime, a rigorous investigation of the similarity of different velocity fields was carried out. The paper is organized as follows. The Simulation method section details the nonlinear DEM model, the computational conditions and parameters, and the statistical method employed to analyze the velocity field. The Results and discussion section presents results from the investigation of the similarity of velocity fields in rotating drums, and discusses the implications of these findings. Finally, conclusions are drawn and future work on the scaling laws of granular materials is summarized in the Conclusions section. Simulation method DEM model The DEM method requires suitable descriptions of the contact mechanism of granular materials. The interparticle interactions are complex to some extent and dissipative in nature owing to inelastic collisions and frictional forces. Additionally, besides the effect of sliding friction, it has been revealed that rolling friction plays an important role, particularly under certain circumstances (Zhu et al., 2007). Among a variety of DEM models for dry particles without adhesion, the formulations proposed by Zhou, Wright, Yang, Xu, and Yu (1999) are adopted in this work; these have been widely employed in various applications (Dong, Wang, & Yu, 2013; Ren et al., 2013). The model is briefly explained below. According to Newton’s second law of motion, particle motion is governed by mi

  dvi = Fnij + Fsij + mi g, dt

(1)

j

and Ii

    dωi ˆi , Ri × Fsij − r Ri Fnij  ω = dt

(2)

j

where mi , vi , ␻i , and Ii are the mass, translational velocity, angular velocity, and moment of inertia of particle i, respectively. The normal and tangential contact forces between neighboring particles i and j are denoted Fnij and Fsij , respectively, while Ri is a vector pointing from the center of particle i to the contact point. The rolling friction coefficient is represented by r , and ␻ ˆ i denotes the unit vector of angular velocity.

Table 1 Parameters used in the present simulations. Parameter

Base value

Particle density,  (kg/m ) Particle diameter, d (mm) Young’s modulus, Y (N/m2 ) Poisson’s ratio () Sliding friction coefficient () Rolling friction coefficient (r ) Normal damping coefficient, n (s−1 ) Drum diameter, D (mm) Drum length, L (mm) Rotational speed, ω (rpm) Filling level (%) Number of particles Time step, t (s) 3

2500 3.0 1.0 × 107 0.5 0.5 0.002 1.0 × 10−6 150 24 32.66 35 6832 4.0 × 10−7

Explored range 1.0–7.0 1.0 × 106 –1.0 × 108 0.1–0.9 0.1–1.0 0.001–0.01 1.0 × 10−7 –1.0 × 10−5 100–210 16–33.6 27.6–40 543–184,608 2.5 × 10−7 –4.0 × 10−7

On the basis of Hertz theory for normal interaction (Johnson, 1985) and Mindlin and Deresiewicz (1953) theory for tangential interaction, the proposed interparticle forces have the form (Zhou et al., 1999) Fnij =

2  3

E

Rı1.5 n − n E

and

 

   R

⎡

Fsij = −sgn ıs |Fnij | ⎣1 −

ˆ ij ın vij · n

 1−



ˆ ij , ]n



min ıs , ıs,max ıs,max

(3)

 1.5 ⎤ ⎦

(4)

where E = Y/(1 −  2 ) with Y and  being the Young’s modulus and Poisson’s ratio, respectively; R¯ = Ri Rj /(Ri + Ri ) with Ri and Rj being ˆ ij is a unit vector pointing the radii of particles i and j, respectively; n from the center of particle j toward the center of particle i; n is the normal damping coefficient accounting for the energy dissipation due to inelastic collisions; and  is the sliding friction coefficient. The parameters ın and ıs are the normal and total tangential displacement, while ıs, max is the maximum tangential displacement given by ıs,max = 

2− ın 2 (1 − )

(5)

This 3D DEM model is implemented in a GPU-based algorithm coded with CUDA (NVIDIA, 2010; Xu, Qi, Ge, & Li, 2011b), and parallelized using the message passing interface (Gropp, Lusk, Doss, & Skjellum, 1996). Owing to the significant speedup of GPU-powered parallel computation provided by the Mole-8.5 supercomputer (Wang & Ge, 2013) at Institute of Process Engineering (IPE), many cases can be simulated for a long period in a systematic parametric study of flow similarity. Simulation conditions As schematically illustrated in Fig. 1(b), the horizontal drum in the present work is partially filled (filling level of 35%) with solid particles. The drum rotates around its axis in the clockwise direction with prescribed angular velocity ω, while the radius and length of drum are denoted R and L, respectively. The geometrical parameters of the drum and particles, and the material properties of the particles are listed in Table 1. The maximum (184,608) and minimum (543) particle numbers correspond to the cases with a drum diameter of 150 mm and particle diameters of 1.0 and 7.0 mm, respectively. Time steps within the range of 2.5 × 10−7 –4.0 × 10−7 s are chosen for different cases according to certain criteria (Li, Xu, & Thornton, 2005; Dong et al., 2013), while the time integration is performed using the popular leap-frog algorithm.

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Fig. 1. Generation of inner particles (a), initial state (b), and cells used in statistical analysis (c) of the 3D rotating drum.

To generate the initial condition for the simulations, a collection of spherical particles is arranged in a square array inside the stationary drum with the particle number depending on the filling level and particle size (see Fig. 1(a)). Subsequently, these particles are allowed to settle under gravity and they eventually reach stable positions with negligible velocities, which are taken as the initial states for the simulations on flow similarity. Additionally, the wall is composed of a large number of “frozen” particles that have the same size and material properties as the inner moveable particles. Since a wall particle is much smaller than the drum diameter in all cases, the effect of the wall particle size on the velocity field is considered to be negligible (Dury, Ristow, Moss, & Nakagawa, 1998). To avoid a lateral wall effect, periodic boundary conditions are applied in the axial direction.

during the sampling period (see the Reference case section). If a cell contains no particle, its velocity is assumed to be zero. As illustrated in Fig. 2, a particle may be partly encompassed by two (a), four (b), or eight (c) cells in 3D space. Theoretically, the volume fraction of a particle in each cell can be calculated by volume integration. For computational convenience, however, the volume fraction is correlated to the particle position and cell center in this study, and different fitting correlations are obtained for these cases. Comparing these fitting results with their theoretical counterparts, errors are found to be less than 0.5% under most conditions. The dissimilarity of a given velocity field from that of the reference case (denoted by the superscript *) is quantitatively characterized according to

Statistical method

P(v) =

The similarities of velocity fields of different size ratios and material properties are investigated under a constant filling level and a Froude number given by

which denotes the ratio of centrifugal to gravitational forces in a rotating drum. As presented in Table 1, the base values of the rotational speed and drum diameter correspond to a Froude number of 8.952 × 10−4 , implying that the granular flows in this work are restricted to the rolling regime (Mellmann, 2001; Van Puyvelde, 2006). Additionally, the rotational speed changes with the drum diameter according to Eq. (6), and this Froude number is therefore maintained in all cases. To facilitate a rigorous comparison of the similarity of velocity fields, a cuboid having dimensions of D × D in the cross-section and L in the axial direction is created to enclose the rotating drum. As shown in Fig. 1(c), the cuboid is further partitioned by a set of orthogonal cells. The velocity vector [v] of each cell is obtained from the velocities of the inner particles (i) following the equation:

where k and N are the cell index and the total number of cells, respectively; ˛ denotes the spatial coordinate; vave and v∗ave are the average scalar velocities of all moving particles in the investigated case and the reference case; while [v]k,˛ and [v]∗k,˛ are the ˛ component of the cell velocity with cell index k in the investigated case and the reference case, respectively. Eq. (8) is in fact similar to the classical definition of the “standard deviation” widely employed in the evaluation of the mixing quality of granular materials (Arratia, Duong, Muzzio, Godbole, & Reynolds, 2006; Ren et al., 2013). When Eq. (8) is applied in practice, the cell should not only be sufficiently small to guarantee high resolution of velocity field but also much larger than the average particle size to include enough particle samples. In the present work, square cells with dimensions of 7.5 mm × 7.5 mm in the cross-section and 12 mm in the axial direction are used, which are 2.5 and 4 times the particle diameter d, respectively; these dimensions are similar to those used in other works (e.g., Liu, Yang, & Yu, 2013). The total number of cells is 800 as shown in Fig. 1(c). Several cell sizes between 1.5d and 3.5d (base value) in the cross-section and between 1.5d and 4d in the axial direction were tested with only insignificant discrepancies found.

[v] =

Results and discussion

Fr =

ω2 R , g

(6)

n v i=1 ( i i )  n i=1

i

(7)

where i and vi are the volume fraction (in the cell) and velocity vector of particle i, respectively. The symbol n is the total number of sampled particles that are completely or partly contained in the cell. Note that the samples are taken from a series of instances



N   k=1

˛

 

[v]k,˛ /vave − [v]∗k,˛ /v∗ave N−1

2 ,

(8)

Reference case The base values of the parameters listed in Table 1 are used in the reference case. After the drum begins rotating, the inner

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Fig. 2. Schematic sketch of the cases when a particle is partly contained in two (a), four (b), and eight (c) cells.

particles gradually gain kinetic energy through their interactions with the rotating wall. Eventually, a slope with a dynamical angle of repose forms, while the particles in the upper region constantly roll down this free surface. The sum of the kinetic energy of all particles is monitored throughout the simulations, and when this value becomes almost constant with only slight fluctuations, it is assumed that the rotating drum has reached the stable state. According to this criterion, a time span of 10–15 s is sampled to obtain the time-averaged velocity field in this study at intervals of 0.02 s. Fig. 3(a) shows a typical snapshot of the rotating drum (crosssection) in the steady state, where the scalar velocities of all particles are distinguished by different colors. In regard to the domain adjacent to the rotating wall, the particle velocity generally decreases as the distance between particle and wall becomes large. Note that a wide central area has minimum velocities of the whole system. Moreover, the velocities of the particles near the free surface are rather high, and some even exceed the velocity of the rotating wall (0.257 m/s). All these phenomena are characteristic of rotating drums in the rolling regime (Mellmann, 2001; Yang, Yu, McElroy, & Bao, 2008; Chou & Lee, 2009). The velocity field corresponding to Fig. 3(a) is further presented by Fig. 3(b), where the magnitude of the velocity in each cell (obtained using Eq. (7)) is also marked by colors. The empty cells are white. It is evident that the cells in the central domain and free surface (particularly its middle part) have relatively low and high velocities, respectively, which is in accordance with the general tendency of the velocity distribution in Fig. 3(a). The maximum velocity in Fig. 3(b) is, however, much lower than that in Fig. 3(a). This difference should be a result of the averaging procedure of the particle velocities inside the cells.

Effect of single parameters As previously discussed in the Introduction section, both the size ratio and material properties of particles would affect the velocity profile in the rotating drums. When the drum diameter remains fixed, the size ratio is uniquely determined by the particle diameter d. The material properties concerned in this work are mainly Young’s modulus Y, Poisson’s ratio , the sliding friction coefficient , the rolling friction coefficient r , and the normal damping coefficient n . The effects of these factors on velocity fields are fully examined over wide ranges, and the dissimilarities between the corresponding velocity fields and the reference case are evaluated through Eq. (8). The dependence of the velocity field of the rotating drum on particle size (within the range of 1.0–7.0 mm) is displayed in Fig. 4. As particle diameter d deviates from its base value (3.0 mm), the dissimilarity of the velocity field P(v) increases rather quickly, revealing the remarkable influence of the particle diameter. Several typical velocity fields are further presented in Fig. 4(a)–(c), corresponding to the cases of d = 1.0, 3.0 (reference case), and 7.0 mm, respectively. The magnitude of the reduced velocity |[v]|/vave is shown by the color of the cell. Note that the discrepancy among these cases is mainly in the central area and on the free surface of the particle layer, and the difference between Fig. 4(a) and (b) is also much larger than that between Fig. 4(b) and (c), confirming the results obtained with Eq. (8) as we compare the corresponding values of P(v). It should also be mentioned that a rather small particle size would result in huge computational cost owing to the large particle quantity, which is already 184,608 in the case of a particle size of 1.0 mm. In view of this limitation and the monotonous relationship between P(v) and particle size,

Fig. 3. Distribution of particles (a) and the velocity field (b) in the steady state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. Dissimilarity of the velocity field versus particle diameter d and the corresponding velocity fields |[v]|/vave for d = 1.0 mm (a), 3.0 mm (b), and 7.0 mm (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

smaller ranges of particle diameter are considered in the following. Fig. 5 presents the effect of sliding friction coefficient  on the velocity field. As  increases or decreases from its base value (0.5), the dissimilarity of the velocity field increases monotonically. When  becomes larger than 0.8 or smaller than 0.3, P(v) already exceeds 0.1, revealing a remarkable effect of this parameter. The sensitivity of the velocity field on  may be ascribed to the fact that the motion of the particles in the drum is mainly driven by the sliding friction force from the rotating wall. The effect of rolling friction coefficient r is shown in Fig. 6. It is found that P(v) slowly increases when r deviates from its base value (0.002). However, P(v) is still lower than 0.1 even when r reaches 0.01 (almost inaccessible in most practical situations). Hence, the rolling friction coefficient has

Fig. 5. Dissimilarity of the velocity field versus sliding friction coefficient.

much weaker influence on the velocity field compared with the sliding friction coefficient. Several typical velocity fields are further presented in Fig. 6(a)–(c), corresponding to the cases with rolling friction coefficients of 0.001, 0.002 (reference case), and 0.01, respectively. Note that the velocity fields of Fig. 6(a) and 6(b) are similar, while there is a large difference between Fig. 6(b) and 6(c), particularly in the central area and on the free surface of the particle layer. The effects of Young’s modulus Y and Poisson’s ratio  are also examined in detail. As presented in the preceding DEM model section, both affect the value of E in Eq. (3). Meanwhile, the Poisson’s ratio also affects the value of ıs, max in Eq. (4) through Eq. (5). As illustrated in Fig. 7, when Y deviates from its base value (1.0 × 107 N/m2 ) in either (positive or negative) direction, the dissimilarity of the velocity field rapidly increases and would eventually exceed 0.1. Furthermore, P(v) grows increasingly more slowly as Y increases from its base value. This is consistent with the common viewpoint that the general characteristics of granular flows in most circumstances are not sensitive to variations in Young’s modulus when it is sufficiently large (typically beyond 1.0 × 109 N/m2 ). In fact, a much smaller Young’s modulus is usually adopted to considerably increase the time step in simulations (Yang, Jayasundara, Yu, & Curry, 2006; Dong et al., 2013). Additionally, the results in Fig. 7 show the large effect of Young’s modulus on the velocity field when it is relatively small (on the order of 107 N/m2 or smaller). It is noted that Poisson’s ratio does not greatly affect the velocity field when it ranges from 0.1 to 0.7. The resulting dissimilarity is lower than 0.1 when the normal damping coefficient ranges from 1.0 × 10−7 to 5 × 10−6 s−1 , implying the comparatively minor effect of this parameter. In summary, for the rotating drum in the rolling regime, changes in the particle size and material properties lead to a certain discrepancy from the velocity field in the reference case. The particle

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Fig. 6. Dissimilarity of the velocity field versus rolling friction coefficient r and the corresponding velocity fields |[v]|/vave for r = 0.001 (a), 0.002 (b), and 0.01 (c).

diameter, sliding friction coefficient, and Young’s modulus have relatively major effects on the velocity fields in the steady state, while the rolling friction coefficient, Poisson’s ratio, and normal damping coefficient have secondary effects in this regard. Effect of multiple parameters To further explore the similarity of velocity fields in the rotating drum, the effects of variations in multiple parameters are investigated as follows. Fig. 8 shows the dissimilarity of velocity fields for different particle diameters (ranging 2.0–4.0 mm) and Poisson’s ratios (ranging 0.1–0.9), where the zone with lower dissimilarity is particularly marked with solid slopes. The two lines, vertical and horizontal, denote the locations where the particle diameter

Fig. 7. Dissimilarity of the velocity field versus Young’s modulus.

d and Poisson’s ratio  equal their respective base values (3.0 mm and 0.5), while their intersection corresponds to the parameters of the reference case. Note that the dissimilarity of the velocity field gradually grows as d deviates from the base value. Additionally, the contours are approximately vertical when  is smaller than 0.8, implying that the particle diameter is more influential than Poisson’s ratio in the determination of the velocity distribution in these areas. However, P(v) rapidly increases with Poisson’s ratio when the ratio exceeds approximately 0.7. In fact, such large Poisson’s ratios are unrealistic for most granular materials. Fig. 9 further illustrates the dissimilarity of velocity fields for different particle diameters and sliding friction coefficients. In contrast to Fig. 8, the contours in Fig. 9 are neither horizontal nor vertical, and thus indicate the comparable effects of these two parameters. Moreover, the zone of the similar velocity field is found to exist only in the domain adjacent to the reference case (d = 3.0 mm and  = 0.5).

Fig. 8. Dissimilarity of the velocity fields for different particle diameters and Poisson’s ratios.

H. Qi et al. / Particuology 22 (2015) 119–127

Fig. 9. Dissimilarity of the velocity fields for different particle diameters and sliding friction coefficients.

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Essentially, the considerable deviation of these two variables from their base values while there is still high similarity with the base case implies a particular relationship between them in determining the steady state of the rotating drum, which should be further analyzed employing the contact model of the DEM method. In general, with increases in the deviation of the size ratio and material properties from their base values, the dissimilarity of the resulting velocity field to the reference case increases. Additionally, when the size ratio varies, an identical velocity field cannot be obtained with suitable adjustment of material properties in the investigated range. Therefore, the size ratio plays a dominant role in the determination of the velocity field. These findings agree well with relevant experimental observations (Orpe & Khakhar, 2001).

Effect of the drum diameter

Fig. 10. Dissimilarity of the velocity fields for different particle diameters and Young’s moduli.

The dissimilarity of velocity fields for different particle diameters and Young’s moduli are plotted in Fig. 10. With an increase in the discrepancy of particle diameter d and Young’s modulus Y from their respective base values, P(v) gradually increases. Additionally, the contours are almost horizontal in the domain close to the base values (d = 3.0 mm and Y = 1.0 × 107 N/m2 ), implying that Young’s modulus has a greater effect than the particle diameter in that region. Moreover, Fig. 11 shows the dissimilarity of velocity fields for different sliding friction coefficients and Young’s moduli. Apart from the domain neighboring the base values ( = 0.5 and Y = 1.0 × 107 N/m2 ), there is a wide area where P(v) is also smaller than 0.1. In that region, the sliding friction coefficient and Young’s modulus even reach 0.8 and 4.0 × 107 N/m2 , respectively.

Fig. 11. Dissimilarity of the velocity fields for different sliding friction coefficients and Young’s moduli.

In contrast to the preceding cases having a fixed drum size, the effect of drum size is further investigated. Although a constant aspect ratio of the drum is not required owing to the application of a periodic boundary condition in the axial direction, the axial length should be sufficiently large to produce reliable results. Additionally, the corresponding number of cells in the axial direction should be guaranteed to be an integer, and reasonable computational cost should also be taken into account. For a rotating drum smaller than that in the base case, a drum diameter of 100 mm and a length of 16 mm is employed, and the rotational speed becomes 40 rpm in this case (constant Froude number). The effects of particle diameter d and sliding friction coefficient  on the velocity field are examined. As presented in Fig. 12(a), the region with minimum dissimilarity (lower than 0.05) is in the domain where d is close to 2.0 mm. Note that the corresponding size ratio of that value is 0.02 (2.0 mm/100 mm), which is identical to the ratio in the base case (3.0 mm/150 mm). The velocity field corresponding to the case of d = 2.0 mm and  = 0.5 (the intersection of the two solid lines in Fig. 12(a)) is displayed in Fig. 12(b). It is seen that Fig. 12(b) is similar to Fig. 4(b) (the reference case) despite small discrepancy in the central area of the particle layer. This further confirms the important role of the size ratio in determining the similarity of velocity fields. Note that the same number of cells as that in the reference case is used in the statistics of velocity fields, while the sizes of cells are linearly scaled in each dimension according to the change in the drum diameter; the cell dimensions are 5 mm in the two directions of the cross-section and 8 mm in the axial direction. Moreover, Fig. 13(a) shows the dissimilarity of velocity fields when the drum diameter increases to 210 mm (with length of 33.6 mm) and the rotational speed becomes 27.6 rpm (constant Froude number). It is found that the values of particle diameter in the domain with relatively small dissimilarity are in the vicinity of 4.2 mm, which is different from the original value of 3.0 mm as shown in Fig. 9. However, the size ratio corresponding to 4.2 mm is 0.02 (4.2 mm/210 mm), which is identical to that of the reference case. The velocity field corresponding to the case with a particle diameter of 4.2 mm and sliding friction coefficient of 0.5 (the intersection of two solid lines in Fig. 13(a)) is further displayed in Fig. 13(b), and is similar to Fig. 4(b) (the reference case). This again confirms that the velocity field of a rotating drum is primarily determined by the size ratio, regardless of the scaling up or down of the drum. Nevertheless, the domain with most similarity (corresponding to dissimilarity lower than 0.05) slightly shifts from the base value of the sliding friction coefficient (0.5) and the expected particle diameter (4.2 mm), revealing the complexity of the scaling behavior of rotating drums.

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Fig. 12. Dissimilarity of the velocity fields for different particle diameters and sliding friction coefficients when the drum diameter is 100 mm (a) and the velocity field |[v]|/vave of the intersection of the two solid lines (b).

Fig. 13. Dissimilarity of the velocity fields for different particle diameters and sliding friction coefficients when the drum diameter is 210 mm (a) and the velocity field |[v]|/vave of the intersection of the two solid lines (b).

Conclusions Under a fixed Froude number corresponding to the rolling regime of rotating drums, the similarity of velocity fields was assessed in DEM simulations for various size ratios and material properties, employing a rigorous statistical method. It was found that both the size ratio and material properties affect the velocity field, but the size ratio has the dominant effect. Additionally, the sliding friction coefficient and Young’s modulus have larger effects than other factors of material properties. Preliminary analysis indicated that the velocity field remains almost unchanged when a constant size ratio is guaranteed for different drum sizes. Generally, the scaling behavior of granular flows is complicated owing to the discrete nature and complex interparticle forces of granular materials. The underlying mechanism of the effects of material properties on scaling behavior should be further explored. The rules of similarity in other flow regimes of rotating drums, as well as granular flows in other circumstances, also deserve systematic investigation.

Acknowledgements This work was supported by the National Key Basic Research Program of China under grant no. 2015CB251402, the National

Natural Science Foundation of China under grant nos. 21206167, 21225628, and 51106168, and the Chinese Academy of Sciences under grant nos. XDA07080203 (the Strategic Priority Research Program) and XXH12503-02-03-03.

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