🎓 Lesson 32
D5
Identifying Fluid Type: Rheogram Interpretation Workshop
A rheogram is a graph that shows how a fluid flows when you push or stir it — like how honey resists pouring versus water.
🎯 Learning Objectives
- ✓ Analyze a given rheogram to classify fluid type using visual and mathematical criteria
- ✓ Calculate apparent viscosity, yield stress, and flow behavior index from rheogram data
- ✓ Explain how rheological properties influence slurry transport in mining hydrometallurgy and tailings management
- ✓ Apply rheogram-derived parameters to select appropriate pump types and pipeline design specifications
📖 Why This Matters
In mining, non-Newtonian fluids — like coal-water slurries, paste tailings, and drilling muds — dominate material handling. Misidentifying their flow behavior leads to catastrophic pump failures, pipeline blockages, and inaccurate dosing in leaching circuits. Interpreting rheograms isn’t academic: it’s the first line of defense against costly operational downtime and safety hazards in slurry transport systems.
📘 Core Principles
Fluids are classified by how shear stress responds to shear rate. Newtonian fluids (e.g., water) show a straight line through the origin (constant viscosity). Non-Newtonians deviate: pseudoplastics (e.g., xanthan gum solutions) thin under shear (concave-down curve); dilatants (e.g., cornstarch–water) thicken (concave-up); Bingham plastics (e.g., bentonite muds) require minimum stress (yield stress) before flowing; and yield-pseudoplastics combine both features. Rheograms reveal these signatures — slope = apparent viscosity, intercept = yield stress, curvature = flow index n. Mastery requires recognizing patterns *before* fitting models.
📐 Power-Law & Bingham Models
The Power-Law model describes pseudoplastic and dilatant fluids; the Bingham model fits yield-stress fluids. Both are derived from rheogram data via linearized log-log or τ–γ̇ plots. Correct model selection prevents erroneous viscosity predictions at operating shear rates.
Power-Law Model
τ = K · γ̇ⁿRelates shear stress to shear rate for time-independent non-Newtonian fluids without yield stress.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| τ | Shear stress | Pa | Force per unit area driving flow |
| K | Consistency index | Pa·sⁿ | Material-specific coefficient reflecting resistance to flow |
| γ̇ | Shear rate | s⁻¹ | Velocity gradient perpendicular to flow direction |
| n | Flow behavior index | dimensionless | Indicator of shear sensitivity: n < 1 = pseudoplastic, n > 1 = dilatant |
Typical Ranges:
Mineral paste tailings (65–75% solids): 0.15 – 0.35
Polymer-thickened leach solutions: 0.4 – 0.7
💡 Worked Example
Problem: A rheometer test on a copper concentrate slurry yields: at γ̇ = 10 s⁻¹, τ = 24 Pa; at γ̇ = 100 s⁻¹, τ = 85 Pa. Determine flow behavior index (n) and consistency index (K).
1.
Step 1: Take log₁₀ of both shear stress and shear rate values: log₁₀(τ₁)=log₁₀(24)≈1.38, log₁₀(γ̇₁)=log₁₀(10)=1.0; log₁₀(τ₂)=log₁₀(85)≈1.93, log₁₀(γ̇₂)=log₁₀(100)=2.0
2.
Step 2: Compute slope n = (log₁₀τ₂ − log₁₀τ₁) / (log₁₀γ̇₂ − log₁₀γ̇₁) = (1.93 − 1.38) / (2.0 − 1.0) = 0.55
3.
Step 3: Solve for K using τ = K·γ̇ⁿ → K = τ / γ̇ⁿ = 24 / (10⁰·⁵⁵) ≈ 24 / 3.55 ≈ 6.76 Pa·sⁿ
Answer:
The result is n = 0.55 (pseudoplastic), K = 6.76 Pa·sⁿ, confirming strong shear-thinning behavior typical of high-solids mineral slurries.
🏗️ Real-World Application
At the Ok Tedi Mine (Papua New Guinea), paste tailings (72% w/w solids, limestone–copper blend) exhibited yield-pseudoplastic behavior with τ_y ≈ 42 Pa and n = 0.28. Engineers used the Herschel–Bulkley model (τ = τ_y + K·γ̇ⁿ) to size positive displacement pumps and specify pipeline velocity > 1.8 m/s to avoid deposit formation — avoiding the 3+ days of unplanned shutdown experienced when earlier Newtonian assumptions were applied to startup flow simulations.
🔧 Interactive Calculator
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