In the Discrete Element Method (DEM), an elongated particle is represented by a sphero-cylinder. Each particle is considered as a discrete entity and its translational and rotational motion in Equations 4, 5 is described by Newton’s second law of motion, m i d v i d t = F i + m i g , (4) I i d ω i d t − I i   · ω i × ω i = M i , (5) where translational and angular velocities of the particle i of the mass m i are represented by v i and ω i , respectively, g is gravitational acceleration, and F i is the resultant contact force exerted on the particle i. The moment of inertia tensor of the particle i is represented by I i and M i is the total torque produced by the contact forces on the particle i .

The resultant contact force between a particle and another particle or a wall boundary can be decomposed into a normal contact force and a tangential one. In this work, the normal contact force F n c follows a linear spring dashpot model, F n c = k n δ n n − C n v n , (6) in which the normal contact stiffness k n in Equation 7 is written as, k n = E i d i · E j d j E i d i + E j d j , (7) and d i and d j denotes diameters of the two contacting sphero-cylinder particles i and j , E i and E j are Young’s moduli, respectively, of the two particles, δ n represents contacting overlap size, and n is the unit vector of the contact normal direction. The second term of the right hand side of Equation 6 accounts for the collisional dissipation of energy, in which C n = 2 β   m * k n is the normal damping coefficient, m * = m i m j m i + m j is the effective mass, and v n is the normal relative velocity vector at the contact point.

Linear spring-dashpot Coulomb limit model (Zhu et al., 2007) is employed to calculate tangential contact force at time t , in Equations 8, 9 expressed as F τ c , t . F τ c , t = min ⁡ F τ , e c , t , μ F n c , t F τ , e c , t F τ , e c , t , (8) in which, F τ , e c , t = F τ c , t − Δ t − K τ v τ Δ t − C τ v τ , (9) and μ is the particle-particle or particle-wall friction coefficient, F n c , t is the normal contact force at time t, F τ c , t − Δ t is the tangential contact force for the previous time step, K τ is the tangential contact stiffness, v τ is the relative tangential velocity vector at the contact point, Δ t is the time step, and C τ is the tangential damping coefficient. When F τ , e c , t < μ F n c , t , it is in static friction stage, otherwise it is in dynamic or sliding friction stage.

The dynamics of elongated sphero-cylinder particles in a rotating drum is simulated. As shown in Figure 1, the particles have a diameter of d = 2 mm and lengths of l p = 4 mm and 10 mm, resulting in the particle aspect ratios of A R = l p / d = 2 and 5, respectively. Monodisperse particles with AR = 5 or binary particle mixtures of AR = 2 and five settle under gravitational forces to densely-packed assemblies in the cylindrical drum of a diameter of D = 100 mm and length of L = 150 mm (Figure 1), before the drum rotates at a specified angular speed about its major axis in the x-direction. The particle properties used for the DEM simulations are listed in Table 1. The choices of the particle properties are based on the biomass materials as reported in (Han et al., 2023).

To examine scale-up rules for the present drum systems, four drums of different diameters, i.e. Drum A to D, are used in the simulations, as shown in Table 2. As the flow patterns and mixing behaviors show no dependences on it, the length of the drum L remains the same for the four drums. To achieve the similarity of the system, the same fill ratio, defined as the total volume of the particles to the volume of the drum, is specified to the four drums, for which the number of particles increases linearly with the volume of the drum. The monodisperse particles have particle aspect ratio of AR = 5. The binary mixtures are composed of two components with AR = 2 and AR = 5, respectively, and both components have volume concentrations of 50% in the mixtures. For the drum systems, a scale-up ratio α is defined as the ratio of cross-sectional area of the current drum ( π D 2 / 4 ) to that of the reference drum. In the present study, Drum A is the reference drum and thus, α spans between 1 and 2.1.

A Froude number F r in Equation 10, reflecting the relative importance of centrifugal and gravitational forces on a particle, can be written as F r = D ω 2 g , (10) in which ω is the rotational speed of the drum and D ω 2 scales with the centrifugal force on a particle, and g is the gravitational acceleration scaling with the gravitational force. Four different Froude numbers are used and corresponding rotational speeds of the drums are listed in Table 3.

Various mixing indices exist and it is critical to choose a proper one to describe the degree of particle mixing for a specific process (Bhalode and Ierapetritou, 2020). The Lacey mixing index (Lacey, 1954) L has been widely used to quantify the mixing degree in various solids handling processes, including in a rotating drum, and it is calculated as, L = S 0 2 − S 2 S 0 2 − S R 2 , (11) in which, S 0 2 = x 1 − x , (12) S R 2 = x 1 − x n , (13) S 2 = 1 N − 1 ∑ i = 1 N x i − x 2 . (14)

In the calculation, the whole particle system is partitioned into many subdomains, namely boxes. The variances of an initial mixing state and a completely mixed state are represented by S 0 2 and S R 2 , respectively, and the current mixing state is S 2 . In Equations 12–14, x is the overall concentration of tracer particles in the whole system, n is the average number of particles in each box, N is the number of the boxes, and x i is the concentration of the tracer particles in the current box i. The lacey mixing index L runs within 0 , 1 , in which 0 represents the completely unmixed state and 1 represents the fully mixed state.

A slope of particles with a wavy surface is formed in the rotating drum, as shown in Figure 2. Larger curvature of the granular surface is obtained for the elongated particles compared to the spherical particles (Cui et al., 2023). To describe the particle flow patterns, the profile of the granular surface is fitted using a straight line. The inclination angle of the straight line θ (Figure 2a) is defined as the dynamic angle of repose, which fluctuates slightly around an average value at the steady flow stage. For a given Froude number Fr, the average dynamic angle of repose is independent on the scale-up ratio α for α ∈ 1 , 2.1 , as shown in Figure 2b. In a drum of a specified scale-up ratio α , the average dynamic angle of repose changes slightly with the Froude number Fr for Fr ≤ 0.5, while a surge of about 10° in the average θ is obtained as Fr increases from 0.5 to 1.1. This sharp increase at F r = 1.10 is attributed to the change in the flow regimes caused by stronger centrifugal effect at a higher rotation speed of the drum. Thus, the average dynamic angle of repose is basically determined by the Froude number for the drums of various sizes.

Different particle flow regimes can be observed in the drums, as shown in Figure 3. The particles have higher velocities as they cascade down on the curved surface. The particles contacting the inner surface of the drum generally have the same velocities to the surface (like non-slip boundary), and thus their velocities increase with increasing Froude number Fr for a given drum. Low-velocity regimes are formed in the centers of the circulating particle assemblies.

The probability distributions of the particle velocities are shown in Figure 4. Low particle velocities correspond to the central region. Intermediate velocities correspond to the boundary layer of the particles contacting the rotating wall, and thus the peak probability occurs at a higher particle velocity in a larger drum. The long tail in a probability distribution curve at the higher particle velocities (particle velocities are greater than 0.2 m/s in Figure 4a and greater than 0.4 m/s in Figure 4b) is associated with the rapid flowing particle surface region, in which shear rates are higher and velocity distribution is wider.

For a given Froude number Fr, the average particle velocity increases with the scale-up ratio α , as shown in Figure 5a, and this velocity increase becomes more significant as Fr increases. A power law relationship is proposed to estimate the dependence of the average particle velocity v a v g on Fr and α , and it is determined by fitting to the data in Figure 5a, v a v g * = v a v g g d p = 2.374   F r 0.4039 α 0.2686 (15) in which v a v g * is a dimensionless average particle velocity expressed as v a v g / g d p . The predictions of Equation 15 are compared with the DEM simulation results in Figure 5b, and a good agreement is observed.

The contact forces exerted on the particles due to particle-particle and particle-wall contacts determine attrition and breakage of the particles and erosion on the drum walls. The average contact force exhibits significant increase as both scale-up ratio α and Froude number Fr increase, as depicted in Figure 6a. In general, higher particle velocities lead to larger contact forces. Thus, the trend of the contact force varying with Fr and α is correlated with that of the average particle velocity (Figure 5a). A power law relationship, which correlates the average contact force F a v g , Fr and α , is determined by fitting to the data in Figure 6a, F a v g * = F a v g ρ g d p 3 = 37.97   F r 0.1383 α 0.3368 (16) in which F a v g * is a dimensionless average contact force expressed as F a v g / ρ g d p 3 . The prediction of Equation 16 is generally consistent with the DEM results, as shown in Figure 6b.

To observe and quantify the mixing behavior, the monodisperse particles are partitioned equally in number and colored differently in blue (left-hand side) and red (right-hand side), as shown in Figure 7. To ensure the consistent particle loading, two rectangular inlet regions with the fixed dimensions were used in all drum. However, as the drum cross-sectional area increases, the relative inlet area becomes smaller, leading to slight variations in the initial packing profile at t = 1 s (Figure 7). In the four drums of different sizes (Drums A-D), similar flow and mixing patterns are observed at the same Froude number of Fr = 0.1007 (Figure 7). Lacey mixing index L, calculated by Equation 11, initially increases with time, then converges to an upper limit, and eventually fluctuates around a constant value for various Fr in different drums, as shown in Figure 8. For a given drum (Figure 8a or Figure 8b), the convergence to the upper limit of the mixing index L occurs earlier at a larger value of Fr, because the higher rotational speed of the drum ω promotes the convective particle flow and thus mixing in the present flow regime in the drum.

According to Figure 8, the Lacey mixing index L and elapsed time t follow an error function (Herman et al., 2022a; He et al., 2024) in the form, L = L f + L 0 − L f exp ⁡ − Rt , (17) in which exp (−) represents the exponential function, L f is the upper limit of the Lacey mixing index, and L 0 is the initial mixing index before the drum rotation. The coefficient R before the time t reflects how fast the material is mixed and it is defined as mixing rate.

Fitting to the Lacey index L-t curves (Figure 8) using Equation 17, the mixing rate R can be determined for each mixing process. As shown in Figure 9a, the mixing rate R is almost independent on the scale-up ratio α at lower values of Froude numbers (Fr = 0.05 ∼ 0.5) while R decreases slightly with α at the larger value of Fr = 1.10. Therefore, when the drums rotate slowly (at small Fr), the mixing rate R is determined by the particle flow pattern and less affected by the drum size; when the drum rotates at a high speed or a larger Fr, the rate of particle mixing is reduced by increasing the drum size or scale-up ratio, since the particles need to move a longer distance for the same mixing outcome in a drum with larger space. In addition, for a given drum, R shows a significant increase with increasing Fr, consistent with the observation in Figure 8, due to the mixing promotion by the higher rotational speed of the drum.

By fitting to the data in Figure 9a, a power law is determined to correlate the mixing rate R, Froude number Fr, and scale-up ratio α as follows, R * = R d p g = 0.0079   F r 0.3084 α − 0.0250 , (18) which indicates that α has a limited impact on the dimensionless mixing rate R * (= R d p / g ) as the magnitude of the exponent above it is very small, and R * is determined mainly by Fr above which the magnitude of the exponent is much larger. A comparison of the scale-up correlation of the mixing rate (Equation 18) and the DEM results is shown in Figure 9b. The accuracy of the prediction of the mixing rate R * based on Equation 18 is less satisfactory than those of the predictions of average particle velocity v a v g * (Figure 5b) and contact force F a v g * (Figure 6b), probably due to the complex nature of the mixing process involving both convection and diffusion of the particle motion, making the remarkable fluctuations in the mixing index L.

For the spherical particles, the average particle velocity V a exhibits a more significant dependence on Fr than α , as the exponent of Fr is much larger than that of α in Equation 1. For the elongated particles, the normalized average particle velocity v a v g * also depends more on Fr than α , while the difference in the exponents of Fr and α in Equation 15 is reduced compared to that for the spherical particles. The total contact force F c t depends more on Fr than α for the spherical particles (Equation 2), while the average contact force F a v g * depends less on Fr than α for the elongated particles (Equation 16). The mixing rate R exhibits a comparable dependence on Fr and α for the spherical particles (Equation 3), while the dimensionless mixing rate R * has a much stronger dependence on Fr than α for the elongated particles (Equation 18).

For the binary mixtures of the elongated particles, the average dynamic angle of repose θ is independent on the scale-up ratio α but shows a significant increase with increasing Froude number Fr, as demonstrated in Figure 10. Such θ -dependence on α and Fr is consistent with that of the monodisperse system (Figure 2B), indicating that the monodisperse and binary systems have the same scale-up behavior for the dynamic angle of repose.

Figure 8a shows the relationship between average particle velocity, scale-up ratio, and Froude number for different drums. It can be observed that the particle velocities improve with the increase in Froude number at a constant drum scale-up ratio. It’s due to the larger speed of the drum because the dominancy of centrifugal forces becomes higher than the inertial forces, and particles are pushed to the drum’s wall easily which enhances the particle’s velocity inside the drum. Furthermore, the average velocity of particles increases with the scale-up ratio, it’s because of drum diameter. Figure 8b shows the comparison plot between DEM simulated velocities and the Fitted velocities of the particles inside the different drums at different Froude number using the following power function:

For the binary mixtures, the average particle velocity v a v g increases with the scale-up ratio α and Froude number Fr (Figure 11a). By fitting to the data in Figure 11a, the power law relationship of the dimensionless average particle velocity v a v g * (= v a v g / g d p ), Fr and α is obtained as, v avg * = 2.398   F r 0.4099 α 0.2569 . (19)

The parameters in Equation 19 are close to those in Equation 15, indicating that the monodisperse and binary systems have the similar scale-up behaviors of the average particle velocities. A very agreement is obtained between the predictions by Equation 19 and the DEM results, as shown in Figure 11b.

As shown in Figure 12a, the average contact force increases with increasing scale-up ratio α at smaller Froude numbers of Fr = 0.05 and 0.1007, while it changes nonmonotonically with α at larger Froude numbers of Fr = 0.50 and 1.10. The interlocking structures are more likely formed in smaller drums (having smaller α ) due to space constraint. Thus, larger contact forces are obtained with α = 1 for Fr = 0.50 and 1.10. The increase of the contact force with increasing α is attributed to the increase in the average particle velocity for α > 1.3. Fitting to the data of Figure 12 gives the power law correlation for the dimensionless contact force F a v g * (= F a v g / ρ g d p 3 ), F avg * = F avg ρ g d p 3 = 35.94   F r 0.2231 α 0.1117 . (20)

Some predictions of the average contact forces by Equation 20 show significant deviations from the DEM results (Figure 12b), as the deviations originate from the non-monotonic changes of the contact force with increasing scale-up ratio at the high Froude numbers.

In the binary mixtures, the longer particles with AR = 5 are colored in red and the shorter particles with AR = 2 in blue. The flow and mixing processes of the binary mixtures in the four different drums are shown in Figure 13. From the perspective of the radial view (the view direction parallel to the axis of the cylindrical drum), similar piling and mixture patterns are observed in the four drums. From the axial view (the view direction perpendicular to the axis of the cylindrical drum), the mixing is uniform in the axial direction of the Drum B, and such uniform mixing in the axial direction is observed in all the four drums. The radial and angular mixing is dominant in the present rotating drums.

In the circulating flows of the binary mixtures, the Lacey mixing index L increases over time before it reaches and fluctuates around an upper limit, as shown in Figure 14. The time duration required to achieve the upper limit decreases with increasing Froude number Fr. It is also observed that the upper limit of L increases with Fr for the binary mixtures (Figure 14), which is different from the monodisperse systems for which the upper limit of L is independent on Fr (Figure 8). The eventual mixing index L of the monodisperse particles is very close to 1, while the eventual L of the binary mixtures at Fr = 0.05 is only about 0.8. Thus, the particle size difference in the binary mixtures prevents the mixing of the two particle components, and the mixing can be improved by increasing Fr (i.e. rotational speed for a given drum).

Like the monodisperse particles (Figure 9a), for the binary mixtures (Figure 15a), the mixing rate R is nearly independent on the scale-up ratio α at lower values of Froude numbers (Fr = 0.05 ∼ 0.5) while R decreases with α at the larger value of Fr = 1.10. By fitting to the data of Figure 15a, the scale-up correlation of the dimensionless mixing rate R * (= R d p / g ) in terms of the Froude number Fr and scale-up ratio in Figure 15b α can be written in a power law relationship, R * = 0.0087   F r 0.1687 α − 0.1115 . (21)

Compared to the correlation of the monodisperse system (Equation 18), the magnitude of the exponent above α in Equation 21 is much larger than that in Equation 18. Hence, the mixing rate of the binary mixtures R has a stronger dependence on the scale-up ratio α than the monodisperse systems.