Ball mills are essential grinding equipment in industries such as mining, cement, and pharmaceuticals, where they facilitate the comminution of materials through the impact and abrasion of grinding media. As a critical component, the liner protects the mill cylinder and influences grinding efficiency by directing the motion of media. However, during operation, the liner undergoes continuous wear due to repetitive impacts and friction from steel balls, leading to increased maintenance costs and reduced mill lifespan. To address this, I employed the Archard wear model within the Rocky-DEM framework to simulate wear patterns under varying liner configurations. My study focused on binary particle systems with 5mm and 10mm media, examining scenarios without liners, different liner shapes (rectangular and trapezoidal), and liner heights (7.5mm, 10mm, and 15mm). This approach allowed me to investigate particle stratification dynamics, energy interactions, and wear mechanisms, aiming to optimize liner design for enhanced durability and efficiency in ball mill operations.
The Archard wear model is widely used for predicting volumetric material loss due to frictional work. The fundamental equation is expressed as:
$$ V = \frac{k F_\tau s_\tau}{H} $$
where \( V \) is the total volume of material worn from the contact surface, \( F_\tau \) is the tangential force applied, \( s_\tau \) is the sliding distance, \( H \) is the hardness of the worn material, and \( k \) is a dimensionless empirical constant. For discrete element method (DEM) simulations in Rocky-DEM, this model is adapted into an incremental form:
$$ \Delta V = C \Delta W_\tau $$
where \( \Delta V \) is the volume worn during a simulation time step, \( \Delta W_\tau \) is the tangential work done by particle collisions, and \( C = k / H \). This formulation enables real-time wear tracking but neglects impact-induced wear, a limitation in my analysis of ball mill systems.
My simulation setup involved a simplified ball mill cylinder model with dimensions 305mm in diameter and 150mm in length, imported into Rocky-DEM. I defined binary particles of 5mm and 10mm steel media with equal quantities to isolate stratification effects. Material properties and interaction parameters were calibrated to mimic real-world conditions, as summarized in Table 1. The ball mill fill rate was determined using:
$$ G = \frac{\pi}{4} D^2 L \gamma \psi $$
where \( G \) is the mass of particles, \( D \) is the mill diameter, \( L \) is the cylinder length, \( \gamma \) is the bulk density (4.8 t/m³), and \( \psi \) is the fill rate (25% in all cases). The rotational speed was set to 50% of the critical speed \( n_c \), calculated as 96 rpm for the cylinder, using:
$$ \phi_n = \frac{n}{n_c} $$
resulting in an operational speed of 47 rpm. This ensured realistic cascading and cataracting motions within the ball mill.
| Parameter | Value |
|---|---|
| Density of cylinder and particles (kg/m³) | 7,800 |
| Poisson’s ratio for cylinder and particles | 0.3 |
| Shear modulus for cylinder and particles (GPa) | 70 |
| Poisson’s ratio for transparent cover | 0.25 |
| Density for transparent cover (kg/m³) | 1,500 |
| Shear modulus for transparent cover (GPa) | 0.198 |
| Particle types | Static friction coefficient | Dynamic friction coefficient | Restitution coefficient |
|---|---|---|---|
| Particle-particle | 0.5 | 0.1 | 0.5 |
| Particle-transparent cover | 0.8 | 0.45 | 0.5 |
Comparative simulations were designed to evaluate liner performance, with cases detailed in Table 2. Wear initiation was set at 20 seconds to exclude unstable particle mixing phases, validated by Lacey index stabilization. Simulations ran for 50-100 seconds, with wear volume loss measured using SpaceClaim software post-simulation.
| Case ID | Liner shape | Liner height (mm) | Number of lifters | Fill rate (%) | Rotation rate (%) | Simulation time (s) |
|---|---|---|---|---|---|---|
| wu | None | 0 | 0 | 25 | 50 | 50 |
| 10-ju-25% | Rectangular | 10 | 12 | 25 | 50 | 100 |
| 10-ti-25% | Trapezoidal | 10 | 12 | 25 | 50 | 100 |
| 7.5-ju-25% | Rectangular | 7.5 | 12 | 25 | 50 | 100 |
| 15-ju-25% | Rectangular | 15 | 12 | 25 | 50 | 100 |
Results revealed distinct particle stratification driven by end-cover effects and lifter capabilities. Without lifters, smaller 5mm particles concentrated centrally due to lower inertia, while larger 10mm particles migrated peripherally, as shown in radial cross-sections. This segregation minimized small-particle collisions with the cylinder, reducing overall wear but compromising grinding efficiency in the ball mill. Introducing lifters altered axial distributions: small particles accumulated at cylinder ends, forming new stratification patterns, whereas large-particle distributions remained stable. Velocity profiles indicated that rectangular lifters promoted more high-speed (7 m/s) cascading motions, increasing effective volume utilization but exposing the liner to direct impacts. In contrast, trapezoidal lifters yielded smoother flows with fewer high-energy collisions, enhancing liner protection.
Energy dissipation and impact spectra were analyzed to correlate collision dynamics with wear. Dissipative energy ranged from \(10^{-17}\) to \(10^{-1}\) J, while impact energy spanned \(10^{-16}\) to \(10^{-2}\) J, with higher frequencies for impacts. Large particles dominated high-energy regions due to mass effects, as described by:
$$ E_k = \frac{1}{2} m v^2 $$
where \( E_k \) is kinetic energy, \( m \) is particle mass, and \( v \) is velocity. Rectangular lifters increased collision frequencies for small particles, elevating cumulative energy transfer. Height variations further influenced outcomes: higher lifters (e.g., 15mm) amplified particle cataracting, reducing “kidney-shaped” inactive zones but intensifying liner impacts. Wear volume measurements confirmed these trends, with rectangular cases showing 15-20% higher loss than trapezoidal ones. However, increasing lifter height gradually decreased frictional wear, as taller lifters confined particles, reducing sliding distances \( s_\tau \) in the Archard model.
Volumetric wear loss data, derived from radial displacement analyses, highlighted critical wear zones. For instance, lifter front faces experienced maximal wear due to counter-clockwise rotation, with no-liner cases exhibiting severe losses. The relationship between wear and dissipative energy was strong, but impact energy correlations were weak, underscoring the Archard model’s limitation in ignoring impact mechanisms. This gap suggests that future ball mill studies should integrate multi-mechanism wear models.
In conclusion, my simulations demonstrate that liner design profoundly affects ball mill efficiency and longevity. Absence of lifters maximizes wear but degrades grinding performance, while rectangular lifters enhance particle agitation at the cost of higher liner degradation. Trapezoidal shapes offer a balance, reducing wear by 10-15% compared to rectangular variants. Increasing lifter height mitigates friction but requires careful optimization to avoid impact-related damage. These insights can guide liner material selection and geometric refinements in industrial ball mills, promoting sustainable operations. Future work should expand to multi-sized media and wet environments to refine predictive accuracy.