🎓 Lesson 15
D5
Power-Law, Bingham, and Carreau Models: Parameter Fitting
These models describe how thick fluids like drilling mud or slurry resist flow — not like water, but in ways that change with how fast you try to move them.
🎯 Learning Objectives
- ✓ Calculate rheological parameters (K, n, τ₀, λ, a) from rotational viscometer data using linearized or nonlinear regression
- ✓ Analyze and select the most appropriate constitutive model for a given fluid based on goodness-of-fit metrics (R², AIC, residual plots)
- ✓ Apply fitted model parameters to predict pressure gradient in pipeline transport of non-Newtonian slurries under field conditions
- ✓ Explain the physical significance of yield stress and flow index in context of slurry handling and pump start-up requirements
- ✓ Design a laboratory test protocol to obtain reliable shear-rate–shear-stress data for model fitting
📖 Why This Matters
In mining operations, fluids like cemented paste backfill, explosive slurries, and tailings transport mixtures behave unpredictably under pumping — they may jam in pipes, segregate in chutes, or require excessive energy to move. Using Newtonian assumptions leads to dangerous underestimation of required pump head or pipe diameter. Mastering these three models lets engineers accurately simulate flow, avoid costly system failures, and comply with safety-critical design standards such as those in CANMET’s Slurry Transport Guidelines or ICMM’s Tailings Management Framework.
📘 Core Principles
All three models describe apparent viscosity ηₐₚₚ as a function of shear rate γ̇. The Power-Law model (ηₐₚₚ = Kγ̇ⁿ⁻¹) captures pure shear-thinning (n < 1) or shear-thickening (n > 1), but cannot represent yield stress. The Bingham model (τ = τ₀ + μₚγ̇) introduces a yield stress τ₀ — the minimum force needed to initiate flow — followed by linear viscous response above that threshold. The Carreau model (ηₐₚₚ = η∞ + (η₀ − η∞)[1 + (λγ̇)²]^(a/2)) bridges realism and tractability: it asymptotically approaches Newtonian viscosity η₀ at low shear, η∞ at high shear, with a transition governed by time constant λ and power-law exponent a. Unlike Power-Law and Bingham, Carreau avoids singularities and better fits real slurry data across wide shear-rate ranges (0.01–100 s⁻¹), making it preferred for CFD-based transport modeling in modern mine design software (e.g., ANSYS Fluent with user-defined rheology).
📐 Key Calculation
Parameter fitting requires transforming raw viscometer data (τ vs. γ̇) into linear or nonlinear forms. For Power-Law, log(τ) = log(K) + n·log(γ̇) yields slope n and intercept log(K). For Bingham, τ vs. γ̇ is linear above τ₀; τ₀ is found via extrapolation to γ̇ = 0. For Carreau, nonlinear least-squares (e.g., Levenberg–Marquardt) is required due to parameter coupling. All fits must be validated with residuals analysis and cross-validation.
💡 Worked Example
Problem: A rotational viscometer returns the following data for a 65% wt. solids tailings slurry: at γ̇ = 1.2 s⁻¹, τ = 4.8 Pa; at γ̇ = 12.5 s⁻¹, τ = 37.2 Pa. Fit a Power-Law model.
1.
Step 1: Take logarithms: log₁₀(τ₁) = log₁₀(4.8) ≈ 0.681, log₁₀(γ̇₁) = log₁₀(1.2) ≈ 0.079; log₁₀(τ₂) = log₁₀(37.2) ≈ 1.571, log₁₀(γ̇₂) = log₁₀(12.5) ≈ 1.097
2.
Step 2: Compute slope n = (1.571 − 0.681) / (1.097 − 0.079) ≈ 0.89 / 1.018 ≈ 0.874
3.
Step 3: Solve for K: K = τ / γ̇ⁿ = 4.8 / (1.2)^0.874 ≈ 4.8 / 1.174 ≈ 4.09 Pa·sⁿ
Answer:
The fitted Power-Law parameters are K ≈ 4.09 Pa·sⁿ and n ≈ 0.87, indicating moderate shear-thinning — consistent with typical fine-particle tailings slurries (n = 0.7–0.9).
🏗️ Real-World Application
At the Antamina Mine (Peru), engineers fitted Carreau parameters (η₀ = 12.4 Pa·s, η∞ = 0.32 Pa·s, λ = 0.85 s, a = 0.41) to paste backfill slurry (82% wt. solids, particle d₅₀ = 22 μm) using a controlled-stress rheometer. These parameters enabled accurate prediction of pipeline friction factor (via Metzner–Reed correlation) and confirmed that startup torque would exceed motor limits unless a progressive-start pump sequence was implemented — preventing repeated tripping during commissioning. The Carreau fit reduced prediction error in pressure drop by 68% compared to Bingham, validating its use in their digital twin hydraulic model.
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