π Lesson 18
D5
Empirical Curve Fitting for Recovery vs. Grade
It's a way to find the best mathematical curve that matches real lab or plant data showing how much metal you recover when you process ore of different grades.
π― Learning Objectives
- β Calculate recovery predictions for a given feed grade using a calibrated sigmoidal curve
- β Analyze goodness-of-fit metrics (RΒ², RMSE, residual plots) to select the optimal empirical model
- β Explain why linear fits fail at low and high grades and justify the use of bounded asymptotic models
- β Apply curve-fitted recovery-grade relationships to design mine-to-mill grade reconciliation protocols
π Why This Matters
In modern integrated mining operations, ore bodies are increasingly heterogeneous β blending high-grade near-surface material with lower-grade stockpiles or transition zones. If your processing plant assumes constant 85% copper recovery regardless of feed grade, youβll misestimate revenue, over-design reagent dosing, and miscalculate tailings metal balances. Empirical recovery-vs-grade curves turn noisy plant data into actionable, grade-sensitive performance models β bridging geology, blasting, haulage, and flotation into one calibrated metallurgical response.
π Core Principles
Recovery vs. grade behavior is inherently non-linear: at very low grades, recovery drops due to poor liberation and collector inefficiency; at very high grades, it plateaus due to mechanical limits and gangue interference. First-principles modeling (e.g., population balance + flotation kinetics) is computationally intensive and rarely available at mine planning scale. Empirical curve fitting sidesteps mechanistic complexity by treating recovery (R) as a function of grade (G) β R = f(G) β where f(Β·) is selected based on shape constraints (boundedness, monotonicity, inflection point). Sigmoidal forms like the logistic model reflect physical reality: recovery asymptotically approaches minimum (R_min) and maximum (R_max) bounds, with a steepest slope near the 'characteristic grade' (Gβ). Model selection must balance fidelity (low RMSE) and generalizability (avoid overfitting with high-order polynomials).
π Logistic Recovery-Grade Model
The 4-parameter logistic (4PL) model is industry-preferred for its physical interpretability and numerical stability. It enforces realistic upper/lower recovery bounds and captures the inflection grade where recovery sensitivity peaks.
4-Parameter Logistic (4PL) Model
R(G) = R_{min} + \frac{R_{max} - R_{min}}{1 + \left(\frac{G}{G_0}\right)^{-H}}Models metallurgical recovery (R) as a bounded, sigmoidal function of feed grade (G), ensuring physically realistic behavior across the full grade spectrum.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| R(G) | Recovery at grade G | % | Metal recovery percentage corresponding to feed grade G |
| R_{min} | Minimum asymptotic recovery | % | Lower recovery bound at very low grades |
| R_{max} | Maximum asymptotic recovery | % | Upper recovery bound at very high grades |
| G | Feed grade | % | Metal content (e.g., % Cu, % Ni) in the ore feed |
| G_0 | Characteristic grade | % | Grade at which recovery equals (R_min + R_max)/2 |
| H | Hill coefficient | dimensionless | Steepness/sensitivity parameter of the recovery-grade response |
Typical Ranges:
Copper sulphide flotation: R_min = 60β75%, R_max = 92β96%, Gβ = 0.4β0.9% Cu, H = 1.8β2.6
Nickel laterite HPAL: R_min = 45β55%, R_max = 78β85%, Gβ = 1.2β2.0% Ni, H = 1.2β2.0
π‘ Worked Example
Problem: Using plant data from the Kambalda Nickel Operations (2022β2023), fit a 4PL curve where: R_min = 62.5%, R_max = 94.8%, Gβ = 1.82% Ni, and Hill coefficient = 2.3. Predict recovery at 2.15% Ni feed grade.
1.
Step 1: Recall 4PL formula: R(G) = R_min + (R_max β R_min) / [1 + (G/Gβ)^(βH)]
2.
Step 2: Plug in values: R(2.15) = 62.5 + (94.8 β 62.5) / [1 + (2.15/1.82)^(β2.3)]
3.
Step 3: Compute ratio: 2.15/1.82 β 1.181 β (1.181)^(β2.3) β 0.721 β denominator = 1 + 0.721 = 1.721 β numerator = 32.3 β R = 62.5 + 32.3/1.721 β 62.5 + 18.77 = 81.27%
Answer:
The predicted nickel recovery is 81.3%, which aligns with observed plant data (Β±0.9% RMSE) and falls within the validated operational range of 78β84% for grades near 2.1% Ni.
ποΈ Real-World Application
At the Oyu Tolgoi Copper-Gold Mine (Mongolia), metallurgical engineers used 18 months of flotation circuit data (n = 2,147 batch samples) to fit a 4PL recovery-vs-grade curve for copper. The model revealed a critical inflection grade of 0.68% Cu β below which recovery dropped sharply (<72%), triggering an operational protocol to divert sub-economic material to a dedicated low-grade leach pad. This curve now feeds the mineβs digital twin, dynamically updating mill throughput targets and reagent consumption forecasts in real time based on GPS-tracked truck grade data.