Introduction & Context

The Herschel‑Bulkley (HB) model describes the shear‑stress response of non‑Newtonian fluids that exhibit a yield stress, a power‑law viscous behavior, and possible shear‑thinning or shear‑thickening. It is expressed as

\[ \tau = \tau_{0} + K\,\dot{\gamma}^{\,n} \]

where τ is the shear stress, τ₀ the yield stress, K the consistency index, n the flow‑behavior index, and γ̇ the shear‑rate. In process engineering the HB model is used to size pumps, design mixers, predict pressure drops in pipelines, and evaluate the energy consumption of processes handling slurries, pastes, polymer solutions, and food products. Accurate parameter fitting enables reliable scale‑up from laboratory rheometry to industrial equipment.

Methodology & Formulas

The fitting procedure follows a linearization of the HB equation after subtracting the known or estimated yield stress:

\[ \tau^{*} = \tau - \tau_{0} \]

Only data points where τ* is positive are retained for regression. Taking natural logarithms yields a linear relationship:

\[ \ln(\tau^{*}) = \ln K + n\,\ln(\dot{\gamma}) \]

Define the transformed variables

\[ x = \ln(\dot{\gamma}), \qquad y = \ln(\tau^{*}) \]

and perform an ordinary least‑squares fit to the straight line y = a + b x. The regression coefficients are obtained from the slope b and intercept a:

\[ n = b, \qquad \ln K = a, \qquad K = e^{a} \]

With the fitted parameters the model predicts shear stress for any shear‑rate:

\[ \tau_{\text{pred}} = \tau_{0} + K\,\dot{\gamma}^{\,n} \]

The goodness‑of‑fit is quantified by the coefficient of determination, \(R^2\), calculated as:

\[ R^{2}=1-\frac{SS_{\text{res}}}{SS_{\text{tot}}} \]

where \(SS_{\text{res}}\) is the sum of squared residuals, defined as \(\displaystyle\sum_{i}\bigl(\tau_{i}-\tau_{\text{pred},i}\bigr)^{2}\), and \(SS_{\text{tot}}\) is the total sum of squares, defined as \(\displaystyle\sum_{i}\bigl(\tau_{i}-\overline{\tau}\bigr)^{2}\). Here, \(\tau_i\) represents the measured shear stress, \(\tau_{\text{pred},i}\) is the shear stress predicted by the fitted model, and \(\overline{\tau}\) is the arithmetic mean of the measured shear stresses.

Interpretation of R²

R² Range	Fit Quality
\(R^{2} \ge 0.90\)	Excellent agreement
\(0.70 \le R^{2} < 0.90\)	Good agreement
\(0.50 \le R^{2} < 0.70\)	Moderate agreement
\(R^{2} < 0.50\)	Poor agreement – model may be inappropriate

Step‑by‑Step Summary

Subtract the estimated yield stress from each measured stress to obtain \(\tau^{*}\).
Discard any data points where \(\tau^{*}\le 0\) (non‑physical for the HB model).
Compute \(x=\ln(\dot{\gamma})\) and \(y=\ln(\tau^{*})\).
Apply linear regression to \((x,y)\) to obtain slope \(n\) and intercept \(\ln K\).
Exponentiate the intercept to recover the consistency index \(K\).
Re‑construct the stress curve using \(\tau_{\text{pred}}=\tau_{0}+K\dot{\gamma}^{\,n}\) and evaluate \(R^{2}\).

The Herschel‑Bulkley model describes the flow behavior of yield‑stress fluids with shear‑thinning or shear‑thickening characteristics. It is expressed as τ = τ₀ + K·γ̇ⁿ, where τ₀ is the yield stress, K the consistency index, and n the flow index. Fitting these parameters allows you to:

Predict pressure drop and pump requirements accurately.
Scale laboratory rheology data to full‑scale process equipment.
Optimize formulation and process conditions for product quality.

To obtain high‑quality data:

Use a calibrated rheometer capable of low‑shear measurements.
Perform a shear‑rate sweep that spans at least three decades (e.g., 0.01–100 s⁻¹).
Allow sufficient equilibration time at each shear rate to reach steady state.
Record both shear stress and shear rate, and repeat measurements to assess reproducibility.

Common approaches include:

Non‑linear least squares (e.g., Levenberg‑Marquardt algorithm) implemented in software such as MATLAB, Python (SciPy), or Excel Solver.
Weighted regression to give more influence to low‑shear data where yield stress dominates.
Initial guesses: τ₀ ≈ minimum measured stress, K ≈ stress at a mid‑range shear rate, n ≈ slope of log(τ‑τ₀) vs. log(γ̇).

Validation steps:

Plot experimental data and the fitted curve on a log‑log scale; the fit should capture both the yield plateau and the power‑law region.
Calculate statistical metrics such as R², RMSE, and confidence intervals for each parameter.
Perform a cross‑validation by fitting a subset of the data and predicting the remainder.
Run a pilot‑scale flow test (e.g., pipe flow) and compare measured pressure drop with model predictions.

Worked Example: Fitting Herschel‑Bulkley Parameters for a Slurry

A process engineer must predict the flow behavior of a mineral slurry that will be pumped through a 150‑mm pipeline at ambient conditions (25 °C, 1 bar). Laboratory rheometry provides shear‑stress versus shear‑rate data, and the engineer uses the Herschel‑Bulkley model to obtain the yield stress (\(\tau_0\)), consistency index (\(K\)), and flow index (\(n\)). The following numerical results have been obtained from a linear regression of the transformed data.

Universal gas constant, \(R = 8.314\;{\rm J/(mol\cdot K)}\)
Temperature, \(T = 25.0\;^{\circ}{\rm C}\)
Pressure, \(P = 1.0\;{\rm bar}\)
Yield stress (initial guess), \(\tau_0 = 3.0\;{\rm Pa}\)
Regression slope (flow index), \(n = 0.709\)
Regression intercept, \(\ln K = 1.544\)
Consistency index, \(K = 4.685\;{\rm Pa\;s^{n}}\)
Sum of squared residuals, \(SS_{\rm res} = 1180.029\)
Total sum of squares, \(SS_{\rm tot} = 26580.222\)
Coefficient of determination, \(R^{2} = 0.956\)

Write the Herschel‑Bulkley equation: \[ \tau = \tau_0 + K\,\dot\gamma^{\,n} \] where \(\tau\) is shear stress (Pa) and \(\dot\gamma\) is shear rate (s\(^{-1}\)).
Linearize the equation by subtracting \(\tau_0\) and taking natural logs: \[ \ln(\tau - \tau_0) = \ln K + n\,\ln\dot\gamma \] This form allows ordinary least‑squares regression on the transformed data.
Perform the regression on the experimental \(\ln(\tau - \tau_0)\) versus \(\ln\dot\gamma\) data. The slope of the best‑fit line gives the flow index \(n\) and the intercept gives \(\ln K\).
Insert the regression results:
- Slope \(n = 0.709\)
- Intercept \(\ln K = 1.544\)
Exponentiate the intercept to obtain the consistency index: \[ K = e^{1.544} = 4.685\;{\rm Pa\;s^{n}} \]
Calculate the goodness‑of‑fit using the sums of squares: \[ R^{2} = 1 - \frac{SS_{\rm res}}{SS_{\rm tot}} = 1 - \frac{1180.029}{26580.222} = 0.956 \] indicating that 95.6 % of the variance in the data is explained by the model.

Final Answer: The fitted Herschel‑Bulkley parameters for the slurry are yield stress \(\tau_0 = 3.0\;{\rm Pa}\), flow index \(n = 0.709\), and consistency index \(K = 4.685\;{\rm Pa\;s^{n}}\). The regression yields a coefficient of determination \(R^{2} = 0.956\), confirming an excellent fit to the experimental data.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle