🎓 Lesson 19
D5
Time-Dependent Liberation Modeling
Time-dependent liberation modeling predicts how long it takes for valuable minerals to become physically free from the surrounding rock during crushing and grinding, based on how the rock breaks down over time.
🎯 Learning Objectives
- ✓ Calculate liberation time constants from laboratory grindability and QEMSCAN®-derived texture data
- ✓ Design comminution circuit residence times to achieve ≥90% target mineral liberation at P80 < 150 µm
- ✓ Analyze the impact of grain size distribution and intergrowth type on liberation kinetics using SEM-MLA datasets
- ✓ Apply time-liberation curves to calibrate flotation recovery models in JKSimMet or METSIM
📖 Why This Matters
In modern integrated mine-to-mill operations, recovering copper from porphyry ores or gold from refractory sulfides hinges not just on how finely you grind—but *when* critical minerals become liberated. Overgrinding wastes energy and generates slimes; undergrinding leaves recoverable metal locked in particles. Time-dependent liberation modeling bridges geology, blasting, crushing, and grinding—ensuring metallurgical recovery targets are met *before* ore hits the flotation bank. It’s the hidden calibration anchor for digital twins in process integration.
📘 Core Principles
Liberation is not instantaneous—it evolves with cumulative specific energy (kWh/t) and residence time in each comminution stage. Three foundational layers govern this: (1) Lithological control—mineral grain size, shape, and intergrowth geometry (e.g., disseminated vs. exsolution textures) set the theoretical minimum liberation size; (2) Mechanical activation—impact, abrasion, and compression induce preferential breakage along grain boundaries, governed by fracture toughness ratios (K<sub>IC,mineral</sub>/K<sub>IC,host</sub>); (3) Kinetic scaling—liberation degree L(t) follows a modified Avrami equation: L(t) = L<sub>∞</sub>[1 − exp(−(t/τ)<sup>n</sup>)], where τ is the characteristic time constant, n reflects breakage mechanism order (n ≈ 0.8–1.3), and L<sub>∞</sub> is asymptotic liberation at infinite time/energy. Calibration requires QEMSCAN® or MLA-derived modal mineralogy coupled with lab-scale timed grinding tests (e.g., SPI® + UG2 liberation assays).
📐 Liberation Kinetics Model
The standard time-dependent liberation model expresses liberation degree L (fraction) as a function of residence time t (minutes) and calibrated parameters. It enables prediction of required grinding time to reach target liberation (e.g., L ≥ 0.92 for chalcopyrite flotation). The model is validated against liberation-by-size assays and embedded in circuit simulation tools.
Avrami-Based Liberation Model
L(t) = L_∞ \left[1 - \exp\left(-\left(\frac{t}{\tau}\right)^n\right)\right]Predicts mineral liberation fraction L(t) as function of residence time t, calibrated time constant τ, kinetic exponent n, and maximum liberation L∞.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| L(t) | Liberation degree at time t | fraction (0–1) | Proportion of target mineral grains fully liberated (exposed surface ≥ 80%) |
| L_∞ | Asymptotic liberation limit | fraction (0–1) | Maximum liberation achievable given ore texture and comminution limits |
| t | Residence time | min | Effective time particles spend in grinding zone (SAG/Ball mill) |
| τ | Characteristic time constant | min | Time required to reach 1 − 1/e ≈ 63% of L∞ under ideal conditions |
| n | Kinetic exponent | dimensionless | Reflects fracture mechanism dominance; determined via timed MLA liberation assays |
Typical Ranges:
Porphyry Cu-Mo (disseminated): 3.5 – 6.2 min
Oxide gold (free-milling): 0.8 – 2.1 min
Refractory sulfide Au (pyrite-hosted): 5.0 – 9.5 min
💡 Worked Example
Problem: A porphyry copper ore has QEMSCAN®-derived chalcopyrite grain size mode = 22 µm, intergrowth factor = 1.4 (disseminated + veinlet), and lab-measured τ = 4.7 min, n = 1.12 at ball mill operating density 75% solids. What residence time achieves 90% chalcopyrite liberation?
1.
Step 1: Identify knowns — L = 0.90, τ = 4.7 min, n = 1.12, L∞ = 0.98 (from MLA assay ceiling)
2.
Step 2: Rearrange Avrami equation: t = τ × [−ln(1 − L/L∞)]^(1/n) = 4.7 × [−ln(1 − 0.90/0.98)]^(1/1.12)
3.
Step 3: Compute: −ln(1 − 0.9184) = −ln(0.0816) ≈ 2.505 → 2.505^(0.8929) ≈ 2.21 → t ≈ 4.7 × 2.21 = 10.4 min
Answer:
The required residence time is 10.4 minutes, which falls within the typical safe range of 8–14 minutes for SAG-Ball circuits targeting P80 = 125 µm.
🏗️ Real-World Application
At Newcrest’s Telfer Mine (Western Australia), time-dependent liberation modeling was used to de-bottleneck the SAG mill circuit. MLA data revealed 30% of pyrite-hosted gold occurred in sub-10 µm grains with high lattice strain. Traditional Bond-based design predicted 11.2 min residence time—but liberation modeling incorporating grain-boundary fracture energy showed L<sub>Au</sub> plateaued at 82% after 9.5 min due to micro-fracture saturation. By adding a pebble crusher and reducing recirculating load, residence time was optimized to 8.7 min—achieving 89% gold liberation while cutting specific energy by 18%. This calibration directly enabled a 4.2% increase in overall gold recovery reported in the 2022 Metallurgical Review (Newcrest Technical Bulletin No. 22-08).