🎓 Lesson 33
D5
Power Law Model Fitting and Apparent Viscosity Prediction
The Power Law model is a simple way to describe how thick fluids like drilling mud or slurry get thinner when stirred faster — like ketchup flowing easier after shaking.
🎯 Learning Objectives
- ✓ Calculate apparent viscosity at any given shear rate using the Power Law parameters K and n
- ✓ Fit Power Law model coefficients (K, n) to experimental rheometer data using log-log linear regression
- ✓ Analyze whether a fluid is shear-thinning, Newtonian, or shear-thickening based on the value of n
- ✓ Apply the Power Law model to predict pressure drop in pipeline transport of non-Newtonian slurries
- ✓ Explain limitations of the Power Law model in low- and high-shear regimes and justify when alternative models (e.g., Herschel-Bulkley) are required
📖 Why This Matters
In mining operations, fluids like tailings slurries, cemented paste backfill (CPB), and ANFO emulsions behave unpredictably under flow — they don’t follow simple water-like rules. Misjudging their resistance to flow can lead to pump failures, pipeline blockages, or incomplete grout placement. The Power Law model provides engineers with a quick, field-deployable tool to quantify this behavior from minimal lab data — turning rheology from abstract science into actionable design input.
📘 Core Principles
All fluids resist deformation via internal friction — quantified as viscosity. Newtonian fluids (e.g., water, oil) have constant viscosity regardless of flow speed. Non-Newtonian fluids — dominant in mining — change viscosity with shear rate. The Power Law isolates this dependency: it assumes the logarithm of shear stress varies linearly with the logarithm of shear rate. The slope (n) reveals fluid type: n = 1 → Newtonian; n < 1 → shear-thinning (most mining slurries); n > 1 → shear-thickening (rare, e.g., cornstarch suspensions). The intercept gives ln(K), reflecting absolute resistance magnitude. While purely empirical and invalid near zero or very high shear rates, it remains the industry’s first-line screening tool due to ease of parameter extraction from rotational rheometer data.
📐 Power Law Shear Stress & Apparent Viscosity
The Power Law defines shear stress and enables calculation of apparent viscosity — the effective viscosity at a specific shear rate. Apparent viscosity (η_app) is not constant but decreases (for n < 1) as flow accelerates — critical for sizing pumps and pipelines.
💡 Worked Example
Problem: A copper mine’s flotation tailings slurry yields rheometer data: at γ̇ = 10 s⁻¹, τ = 8.5 Pa; at γ̇ = 100 s⁻¹, τ = 42.3 Pa. Fit the Power Law model and compute η_app at γ̇ = 50 s⁻¹.
1.
Step 1: Take logs: ln(τ₁) = ln(8.5) ≈ 2.14, ln(γ̇₁) = ln(10) ≈ 2.30; ln(τ₂) = ln(42.3) ≈ 3.75, ln(γ̇₂) = ln(100) ≈ 4.61
2.
Step 2: Compute slope n = (ln τ₂ − ln τ₁) / (ln γ̇₂ − ln γ̇₁) = (3.75 − 2.14) / (4.61 − 2.30) ≈ 1.61 / 2.31 ≈ 0.697
3.
Step 3: Solve for K: K = τ / γ̇ⁿ = 8.5 / (10^0.697) ≈ 8.5 / 5.0 ≈ 1.70 Pa·sⁿ
4.
Step 4: Compute η_app at γ̇ = 50 s⁻¹: η_app = K γ̇^(n−1) = 1.70 × 50^(−0.303) ≈ 1.70 × 0.573 ≈ 0.974 Pa·s
Answer:
The apparent viscosity at 50 s⁻¹ is 0.97 Pa·s, consistent with shear-thinning behavior (n = 0.70 < 1).
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), operators transport ~35% w/w copper tailings slurry through a 12-km, 300-mm-diameter pipeline. Initial pump sizing using Newtonian assumptions led to 22% higher energy consumption than predicted. After fitting Power Law parameters (K = 2.4 Pa·sⁿ, n = 0.58) from onsite rheometry, revised hydraulic calculations reduced predicted head loss by 18% and enabled downsizing of motor drives — saving $140k/year in electricity. The model also guided optimal pipeline velocity (1.8–2.2 m/s) to avoid deposition while minimizing erosion.
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