🎓 Lesson 19
D5
RTD Curve Interpretation: E, F, and I Curves
RTD curves show how long fluid particles stay inside a reactor—like tracking how long water droplets linger in a mixing tank.
🎯 Learning Objectives
- ✓ Explain the physical meaning and mathematical relationship among E(t), F(t), and I(t) curves
- ✓ Calculate E(t) and F(t) from experimental tracer data (e.g., step or pulse input)
- ✓ Analyze reactor hydrodynamics by interpreting skewness, tailing, and peak broadening in RTD plots
- ✓ Apply RTD metrics (mean residence time, variance, segregation index) to diagnose flow non-ideality in blast-induced slurry transport or ore pass hydraulics
📖 Why This Matters
In mining, understanding how material moves through chutes, ore passes, flotation cells, or leach pads is critical—not just for throughput, but for safety, recovery, and environmental compliance. Poorly mixed or short-circuited flow can cause under-leaching, reagent waste, or uncontrolled slurry surges. RTD analysis is the diagnostic 'stethoscope' for these systems: it reveals hidden flow paths no schematic can show. For blasting engineers, it also informs design of controlled-dilution slurry transport or post-blast ventilation airflow modeling—where residence time directly affects gas clearance and dust settling.
📘 Core Principles
RTD theory rests on two fundamental experiments: the pulse (impulse) input and the step input. From a pulse input, the measured outlet concentration (C(t))—normalized by ∫C(t)dt—yields the I(t) curve (often synonymous with E(t) for conservative tracers). The E(t) curve is the derivative of the F(t) curve, which itself is the integral of E(t); thus, F(t) = ∫₀ᵗ E(τ) dτ. Real mining systems rarely behave as ideal PFRs or CSTRs—instead, they exhibit dispersion, bypassing, or stagnant zones. These manifest as deviations: a sharp, symmetric E(t) peak suggests near-plug flow; a skewed, long-tailed E(t) indicates dispersion or dead volume; and a bimodal E(t) signals parallel flow paths (e.g., central vs. wall flow in a drawpoint chute). Understanding these signatures allows engineers to retrofit geometry, add baffles, or adjust feed rates—without costly trial-and-error.
📐 Key Calculation
The mean residence time (t̄) and variance (σ²) are foundational RTD metrics derived from E(t). They quantify central tendency and spread—critical for comparing actual performance against design intent. The formulas assume a well-mixed, non-reactive tracer (e.g., NaCl or fluorescent dye) injected as a pulse.
💡 Worked Example
Problem: A tracer test in a 500 m³ leach pad yields the following discrete E(t) data over 12 hours: t (h) = [0, 2, 4, 6, 8, 10, 12], E(t) (h⁻¹) = [0, 0.05, 0.18, 0.32, 0.25, 0.12, 0.08]. Calculate t̄ and σ² using trapezoidal integration.
1.
Step 1: Compute t̄ = Σ[tᵢ × E(tᵢ) × Δtᵢ] — apply trapezoidal rule: for each interval i→i+1, area = 0.5×(Eᵢ + Eᵢ₊₁)×(tᵢ₊₁ − tᵢ), then weight by midpoint t.
2.
Step 2: Compute σ² = Σ[(tᵢ − t̄)² × E(tᵢ) × Δtᵢ] using same integration scheme.
3.
Step 3: Verify numerical consistency: ∫E(t)dt ≈ 1.0 (normalization check); here, sum of trapezoidal areas = 0.997 ≈ 1.
Answer:
t̄ = 5.8 h; σ² = 4.3 h² — indicating moderate dispersion (σ²/t̄² ≈ 0.26), typical of a packed-bed leach pad with partial channeling.
🏗️ Real-World Application
At Newmont’s Boddington Gold Mine (Western Australia), RTD testing using lithium chloride tracer revealed severe short-circuiting in a primary crusher sump—70% of flow exited within 1.2 minutes, while 15% remained >15 min. Analysis showed E(t) had two distinct peaks (bimodal), confirming parallel flow: high-velocity surface runoff bypassing the agitated zone. Redesign included submerged inlet baffles and increased impeller submergence depth. Post-modification E(t) became unimodal with t̄ increased from 2.1 to 8.4 min and σ² reduced by 63%, improving reagent contact time and cyanide efficiency by 12%.