1. Introduction and Objectives
Cohesive granular materials are encountered in a variety of industries, such as food processing, agriculture, geotechnical industries, and pharmaceuticals. These materials go through a range of processing operations, from feeding [
1], mixing [
2], granulation [
3,
4], drying [
5], roller compaction [
6] to transport and storage operations [
7]. The flow of these materials can influence product quality, production rate, and the type of manufacturing operations utilized in product design [
8]. This makes the study of cohesive granular flow an essential area of research.
The discrete-element method (DEM) has become a popular computational technique for simulating granular motion ever since Cundall and Strack [
9] first demonstrated its usefulness for studying granular behavior. The movement of particles in DEM is governed by the laws of physics, and DEM tracks every particle and all its interactions throughout a simulation domain. It can be used to study the effects of changes in material properties, particle sizes, equipment design, and operating conditions [
10], making it a useful tool to gain insight related to the flow of particles in equipment.
DEM can be used to simulate cohesive granular materials with the help of cohesive contact models. Cohesiveness can be due to the inherent nature of a material or it can be caused by wetting with water or other binding agents. Different contact models exist in the DEM literature for simulating different types of cohesive behaviors, such as the linear cohesion model [
11], the JKR model [
12], and liquid-bridge models [
13,
14]. These models have been shown to be reliable for prediction in many studies. The JKR model was used in a study of mixing of glass beads in a single blade mixer [
15]. Changes in mixing performance were found to be similar for both dry and wet glass beads for the same changes in the blade rake angles. Mukherjee et al. [
16] studied the discharge rates of cohesive particle flow in a hopper by using a contact model containing a dimensionless parameter called the Bond number that combined the cohesion effects of van der Waals forces, electrostatic forces, and force due to liquid bridges. The study found that the hopper discharge rates decreased with increasing Bond numbers and also that there would be no discharge above a critical Bond number as the cohesive forces were more dominant than the particle weight. Guo et al. developed a DEM model to predict the flow behavior of wet fibers with the help of a liquid-bridge contact model and validated it by comparing the simulated angle of repose values to experimental values [
17]. Radl et al. used a capillary force model to simulate wet granules inside a four-bladed mixer and showed that the mixing of particles with low moisture is heterogeneous due to agglomerates that break at specific regions near the impeller [
18]. A study by Wu et al. [
19] simulated the aggregate skeletons of different gradations in asphalt mixture and found that there is higher load resistance in fine-gradation compared to coarse-gradation of asphalt mixture. Remy et al. [
20] used a DEM study to find that mixing performance improves for granular particles with low moisture content when compared to dry material, but the performance deteriorates when the moisture content is high.
Although DEM is good for simulating granular flow behavior, it has three major limitations:
DEM simulations are computationally expensive and take a long time to complete due to small time-step sizes and large numbers of particles that are tracked as they move through the system.
The approximation of complex particle shapes using spheres, usually done to reduce computational burden, might not reflect the true nature of the particles being simulated.
Calibration of DEM parameters to match particle flow behavior in the experimental system is a challenge.
The first two limitations can be reasonably addressed if one has access to high-performance computing resources [
21]. We seek to address the third limitation in this study.
Some of the methods available in the literature for the calibration of DEM parameters for noncohesive systems are described here. Sandpile tests were used to adequately calibrate the rolling-friction coefficient between particles and between a particle and wall [
22]. Sen et al. [
23] studied the mixing of three free-flowing materials in a bin blender by first calibrating the friction coefficients of those materials by using angle-of-repose simulations. The mixing model was then used to identify superpotent and subpotent regions inside the blender. Zhou et al. [
24] also conducted angle-of-repose simulations on noncohesive spheres and formulated a calibrating equation relating the angle of repose to coefficients of sliding and rolling frictions. Do et al. [
25] used genetic-algorithm and direct-optimization techniques to calibrate coefficients of sliding and rolling frictions of the DEM particles based on experiment results of angle-of-repose tests and hourglass discharge times for quartz sand. Simons et al. [
26] conducted sensitivity analysis on ring shear-cell DEM simulations for cohesionless particles and concluded that sliding- and rolling-friction parameters can be calibrated using the shear cell if Young’s modulus was predefined. Cheng et al. [
27] developed an iterative Bayesian tool for the calibration of the shear modulus and the friction coefficients of glass beads that underwent loading and unloading cycles of oedometric compression. A study by Ostanin et al. [
28] calibrated the parameters for a mesoscopic DEM simulation of nanocrystalline particles by matching peak force and critical strain of nanoindentation tests in the simulations. A method for calibrating DEM parameters to simulate cohesionless soil using in situ sinkage tests and nonlinear optimization was developed by Asaf et al. [
29].
Although a large variety of cohesion models exist, the methods available for calibrating cohesive systems are far fewer compared to free-flowing systems. The angle of repose method, which works well for noncohesive systems, might not apply to cohesive systems, as cohesive forces might be too large for gravitational flow. Tsunazawa et al. [
30] developed a cohesion model that considered liquid-bridge and adhesion forces, and validated the cohesive parameters by showing that the flow of wet glass beads in a pan pelletizer was in good agreement with the experimental flow behavior at various rotation speeds. In this study, the sliding-friction parameter was calibrated using angle-of-repose experiment values for the dry material. Wang et al. [
31] calibrated the linear viscous coefficient and the local damping parameter for a mesoscopic DEM model of two carbon nanotubes using data collected from molecular-dynamics simulations. A DEM study of the FT4 powder rheometer on silanized glass beads [
32] utilized an experimental drop test developed by Zafar et al. [
33] to determine the surface-energy parameter for the JKR cohesive model.
This work proposes an alternative method for the calibration of highly cohesive particles by exploring the use of dynamic yield strength (DYS) and shear-cell simulations as calibration methods for simulating wet granular particles using the JKR cohesion model in DEM. The objectives of this study are as follows:
Develop a DEM model for dynamic-yield-strength measurements and use it to calibrate the surface energy between particles.
Develop a DEM model for a parallel-plate shear cell and calibrate the coefficient of sliding friction (CoSF) and the coefficient of rolling friction (CoRF) parameters for particle–particle interactions.
3. Method and Simulation Setups
All simulations were performed using EDEM
® version 2.7.3 (DEM Solutions) [
34]. The geometry for the shear-cell simulation was developed using the 3D-CAD package called SOLIDWORKS
® (Dassault Systémes) [
35], while the geometry for the DYS simulation was developed using tools in EDEM
®.
In a DEM model, particles are inserted into the simulation domain, and contacts between particles and particles, and between particles and boundaries, are detected. Since the soft-particle DEM approach was used here, particle overlaps were calculated next. Next, the force on each particle is determined. Based on this force, particle acceleration is calculated using Newton’s laws of motion. The acceleration values for each particle are integrated over a small time step in order to determine the velocity and position of the particles for the next time step [
10].
The total contact force (
) acting on each particle is the sum of the normal contact force (
), the normal damping force (
), the tangential contact force (
), the tangential damping force (
), and the body force (
) (
1). The only body force considered in the DEM models in this study is the force due to gravity.
The particles in the experiments were cohesive, wet, and in the funicular region. In order to calculate the forces that arise due to particle contacts (with other particles and boundaries), DEM uses a physics/contact model. The capillary force contact model was not applicable as the experiments were not conducted in the pendular region. Since a viscous force model was not available in EDEM
®, the contact and cohesive forces in the DEM models developed in this study were also accounted for by using the JKR normal contact model [
12]. This contact model uses a parameter called surface energy (
) to quantify the attractive nature of the particles. When surface energy is set to zero, the JKR model reduces to the Hertz contact model, a nonlinear spring-and-dashpot model based on the Hertz theory of elastic contacts [
36]. The normal contact force between two spherical particles of radius
and
with a normal overlap,
is given by [
37]:
where
a is the contact radius between the particles,
is the equivalent radius, i.e., the inverse of the reciprocal sum of the two particles’ radii
and
, and
is the equivalent Young’s modulus, i.e., the inverse of the reciprocal sum of the two particles’ Young’s moduli
and
. Young’s modulus is calculated from the shear modulus (
G) and Poisson’s ratio (
) using the relation:
= 2
(1+
).
The normal damping force is related to the normal stiffness (
), relative normal velocity (
), equivalent mass (
), and coefficient of restitution (e) by:
Tangential forces are calculated using the Mindlin–Deresiewicz no-slip model [
38]. The tangential contact force and the tangential damping force are given by the following equations:
where,
is the tangential stiffness,
is the equivalent shear modulus,
is the tangential overlap, and
is the relative tangential velocity. Coulomb’s law limits the tangential force by the coefficient of static friction (
), as given by [
39]:
The rolling torque (
) on a particle is calculated using the rolling-friction resistance model given by the equation [
24]:
where
is the rolling friction coefficient, and
is the angular velocity.
3.1. Dynamic-Yield-Strength Simulation Setup
Figure 2 shows the the DEM model setup for the DYS simulations. The platens were simulated by constructing two solid cylindrical plates of 50 mm diameter. Just as in the experiments, the top platen was movable while the bottom platen remained fixed throughout the simulation.
In order to make the pellet in the simulation, a hollow cylindrical die of 25 mm diameter and 40 mm height was created with both its top and bottom circular sections open. This cylinder was placed vertically over the bottom platen and was filled to a height of 29 mm with spherical particles, which were normally distributed, with a mean diameter of 0.7 mm and a standard deviation of 0.21 mm. Particle distribution was capped at 0.7 and 1.0 mm to simulate the sieve fraction from the experiment results. The spherical particles in the simulation represent an approximation to the experimental granules, which had a vertical aspect ratio of 1.30. This increase in size is a common approximation done in DEM simulations in order to reduce computational time [
40]. These particles were compressed inside the die at a rate of 1.5 mm/s to a height of 22 mm using a smaller cylindrical plate of 24.6 mm diameter, and then released. This caused the pile to spring back up to an approximate height of 25 mm. The hollow cylinder was then removed to leave behind a cylindrical pellet on top of the bottom platen.
The dynamic-yield-strength test was carried out by moving the top platen vertically downwards at a speed of 100 mm/s. The force on the top of the pellet and the deformation it underwent were recorded. A plot of the engineering stress–strain curve was made, and peak-stress value gave dynamic yield strength.
The details of the DEM parameters used in the DYS simulations are presented in
Table 2 and
Table 3. The majority of the parameters used in the base-case simulation were taken from the literature [
23]. The density of the particles was set to the experimentally measured density of the granules. Since the platens were lubricated in the experiment, they were assumed to be frictionless in the simulations, and the sliding- and rolling-friction parameters between a particle and wall were set to zero. Additionally, the value of surface energy was chosen by running test simulations to determine the minimum value that was required to keep the pile of particles stable after the precompression step. This value was doubled and used as the base-case setting.
3.2. Periodic Shear-Cell Simulation Setup
A DEM simulation of the full experimental shear cell was found to be too computationally expensive. Hence, a periodic section of the shear cell, similar to the one developed by Ketterhagen et al. [
41], was used for the simulations. The periodic-section simulation had the dimensions of 18 × 23 × 22 mm in the x, y, and z directions, respectively. The x-axis was treated as a periodic boundary. The top plate consisted of a horizontal plate of 18 × 22 mm with grooves of height 3 mm and width 0.8 mm, with a spacing of 10.5 mm. It also had two vertical plates of dimensions 18 × 8 mm on either side. The bottom plate consisted of a horizontal plate of 200 × 22 mm with grooves of height 2 mm and width 1 mm, with a spacing of 5.35 mm. Just like the top plate, the bottom plate also had two vertical plates of dimensions 200 × 14.5 mm on either side.
Figure 3 shows the shear-cell simulation when empty. In the simulations, the particles were periodic but the bottom plate was not, since the boundary elements could not be made periodic due to software restrictions. This should not affect the results from the simulation since the bottom plate had grooves that repeated at regular intervals, and as this plate moved as a single unit at a constant speed causing the particles close to the plate to be subjected to the same constant force, even when the particles moved across the periodic boundaries.
The shear cell was simulated with 20,000 particles to match the experimental compressed bed height of 16 mm. The top plate was moved down on the particles to apply a constant stress of 2000 Pa in the preconsolidation step. The application of constant load was achieved by using the Dynamics Coupling feature in EDEM®. Once the particles were compressed by the top plate, the bottom plate was made to move at a rate of 3.813e-05 m/s in the positive x-direction. This speed is the experimental angular velocity converted to linear speed. The shear stress on the top plate was recorded.