Reference ID: MET-96A7 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Bingham fluids exhibit a finite yield stress τ0 that must be exceeded before flow begins. Once yielded, the shear stress increases linearly with shear rate, giving a constant plastic viscosity μp. Accurate determination of τ0 and μp is essential for sizing pumps, pipes, mixers, and extruders in food, pharmaceutical, mineral, and wastewater processes. The simplest and most widely accepted method is linear regression of steady-state rheogram data collected in the laminar regime.
Methodology & Formulas
Data acquisition
Measure steady shear stress τ at several shear rates γ̇ in the laminar range. Ensure the instrument gap and geometry satisfy the following limits:
Regime
Condition
Typical threshold
Laminar (Couette)
Re = ρ ω R δ / μp
Re < 1700
No slip at wall
roughness < 5 % gap
—
Linearisation
The Bingham model is
\[ \tau = \tau_0 + \mu_{\text{p}} \dot{\gamma} \]
which is linear in γ̇. Treat τ as the dependent variable and γ̇ as the independent variable.
Regression coefficients
For n data pairs (γ̇i, τi) compute the sums
\[
S_1 = \sum_{i=1}^{n} \dot{\gamma}_i,\quad
S_2 = \sum_{i=1}^{n} \dot{\gamma}_i^{2},\quad
S_3 = \sum_{i=1}^{n} \tau_i,\quad
S_4 = \sum_{i=1}^{n} \dot{\gamma}_i \tau_i
\]
The slope (plastic viscosity) and intercept (yield stress) are
\[
\mu_{\text{p}} = \frac{n S_4 - S_1 S_3}{n S_2 - S_1^{2}},\qquad
\tau_0 = \frac{S_3 - \mu_{\text{p}} S_1}{n}
\]
Unit conversion
Plastic viscosity is often reported in centipoise: μp,cP = μp / 10−3
Yield stress may be quoted in bar: τ0,bar = τ0 / 105
Collect steady-state shear-stress (τ) data at increasing shear rates (γ̇).
Fit the linear portion of the curve (high-γ̇ region) to the Bingham model: τ = τy + μp γ̇.
Extrapolate the straight line to γ̇ = 0; the intercept is the yield stress τy.
Discard data below the transition shear rate where the curve deviates from linearity; these points represent material creep before yielding.
Controlled-stress rheometer with vane geometry: minimizes wall slip and measures true τy within ±2 %.
Stress-growth test: apply a constant low stress and record the strain; the stress at which strain suddenly increases is τy.
Slump test (for high-concentration pastes): correlate slump height to τy via calibrated empirical equations; accuracy ±5 %.
No. Single-point instruments assume a Newtonian response; Bingham fluids require at least two data points to separate τy and plastic viscosity μp. Without the intercept, the yield stress is indeterminate.
Typical k values for mineral slurries: 0.02–0.04 °C−1; a 20 °C rise can halve τy.
Always measure τy at the process temperature or apply the Arrhenius correction before scale-up.
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Worked Example – Yield Stress of a Drilling Mud
A process engineer is characterising a new water-based drilling mud that must circulate through a 150 mm ID riser. Laboratory rheometer data (shear rate \(\gamma\) vs. shear stress \(\tau\)) were collected at 25 °C and need to be fitted to the Bingham model so that the pump specification can be completed.
Knowns
Number of data points, n = 4
\(\sum\gamma\) = 65 s⁻¹
\(\sum\tau\) = 406 Pa
\(\sum\gamma^2\) = 1425 s⁻²
\(\sum\gamma\tau\) = 8220 Pa·s⁻¹
Step-by-Step Calculation
Compute the denominator for the linear-regression slope:
\[
\text{denom} = n\sum\gamma^2 - (\sum\gamma)^2 = 4(1425) - (65)^2 = 1475
\]
